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



<b>assume </b><a NAME="E1:31"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Olive">i</font> is <a href="pencil_1.html#V10">strongly_connected</a>
 ;<br/>

<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E1:31_1"/>
<font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">o</font> is <a href="pencil_1.html#V10">strongly_connected</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:31_1"/>
<font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">o</font> is <a href="pencil_1.html#V9">connected</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>let </b><font color="Maroon">f</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">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">b1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>, <font color="Maroon">b2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E3:31"/><i><font color="Green">E37</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">B1</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>, <font color="Maroon">B2</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2"/>
( <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font> or <font color="Maroon">B1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>suppose </b></a><a NAME="E1:31_2_1"/>
<font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">D</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E3:31_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i>, , </i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> implies for <font color="Olive">D1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">d1</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st <font color="Olive">D1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">d1</font> holds <br/> <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Olive">d1</font> );<br/><a NAME="E4:31_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:31_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:31_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:31_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:31_2_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">D2</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E4:31_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">d2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E5:31_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">d2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_2_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_1_1_1_1"/>
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">d2</font>
 <b>by </b><i>, <a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_2_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:31"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">D1</font> = <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">d1</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:31_2_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">d1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">d1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">d1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/><a NAME="E6:31_2_1_1_1_1_2"/>
( <font color="Maroon">D1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">D2</font> )
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E3:31"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E7:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">d1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">d2</font>
 <b>by </b><i>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/><b>thus </b><a NAME="E8:31_2_1_1_1_1_2"/>
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">d2</font>
 <b>by </b><i>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:31_2_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="txt" href="pencil_3.html#E1:31"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:31"><i><font color="Green">E35</font></i></a>);<br/></i><a NAME="E7:31_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>]
 ;<br/><a NAME="E8:31_2_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E9:31_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:31_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:31_2_1"/>
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_2"><b>suppose </b></a><a NAME="E1:31_2_2"/>
<font color="Maroon">B1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B2</font>
 ;<br/><div class="add"><a NAME="E2:31_2_2"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:31_2_2"/>
 <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="funct_2.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
