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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> is <a href="pencil_1.html#V10">strongly_connected</a>
 ;<br/>

<div><i><font color="Green">E26</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">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:26_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V10">strongly_connected</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pencil_1.html#V9">connected</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pencil_2.html#NM2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E3:26"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E4:26"/><i><font color="Green">E28</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></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">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></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">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:26_2_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_3.html#E4:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pencil_3.html#T23" target="_self">Th23</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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">c<sub>2</sub></font></span>)</span> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:26_2_1"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E6:26_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E7:26_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:26_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E9:26_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E10:26_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E11:26_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E12:26_2_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:26_2_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:26_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E5:26_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:26_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:26_2_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">b<sub>3</sub></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">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_2_1_2"/><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E2:26_2_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E3:26_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:26_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:26_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:26_2_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</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">E31</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E8:26_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E10:26_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E11:26_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E13:26_2_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E14:26_2_1_2"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="funct_2.html#V3">bijective</a> <a href="funct_2.html#NM1">Function</a> of 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">c<sub>2</sub></font></span>)</span>,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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:26_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:26_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E7:26_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T125">FUNCT_1:125</a></i>;<br/>

<b>thus </b><a NAME="E16:26_2_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:26_2_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="pencil_3.html#E7:26_2_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pencil_3.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/><a NAME="E15:26_2_1"/>
<font color="Maroon">c<sub>3</sub></font> is  <a href="funct_2.html#V3">bijective</a> <a href="funct_2.html#NM1">Function</a> of 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">c<sub>2</sub></font></span>)</span>,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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/><a NAME="E16:26_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_1.html#T125">FUNCT_1:125</a></i>;<br/><b>hence </b><a NAME="E17:26_2_1"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_3.html#E8:26_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="pencil_3.html#E11:26_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="pencil_3.html#E12:26_2_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_3.html#T23" target="_self">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">c<sub>4</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:26_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="txt" href="pencil_3.html#E4:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pencil_3.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E3:26_2_2"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
