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<div class="add">

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



<b>assume </b><a NAME="E1:33"/><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/>

<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>set </b><font color="Maroon">s</font> =  <a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font>;<br/>
<b>take </b>
 <a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>  st <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_3.html#K4" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Olive">i</font>;<br/>
<a NAME="E2:33"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E3:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i1</font>, <a href="pencil_3.html#K4" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>]
 
;<br/>
<b>hence </b><a NAME="E4:33_1"/>
 ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">j</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">B</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:33_2"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="pencil_3.html#K4" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:33_2"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as  <a href="funcop_1.html#V1">Function-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E7:33"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_3.html#K4" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:33"/>
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>  st <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Olive">m</font> is <a href="pencil_2.html#V1">isomorphic</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_3"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:33"><i><font color="Green">E34</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E2:33_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>consider </b><font color="Maroon">L</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="E4:33_3"/><i><font color="Green">E49</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">L</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:33_3"/>
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pencil_3.html#K3" target="_self">canonical_embedding</a> <font color="Maroon">f</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E1:33"><i><font color="Green">E34</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E6:33_3"/>
<font color="Maroon">m</font> is <a href="pencil_2.html#V1">isomorphic</a>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:33"><i><font color="Green">E35</font></i></a>, , </i>;<br/>


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

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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">a</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E9:33"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><a NAME="E10:33"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E11:33"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>consider </b><font color="Maroon">b1</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="E13:33"/><i><font color="Green">E55</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <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; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:33"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E14:33"/>
<font color="Maroon">m</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">b1</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E1:33"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E4:33"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E15:33"/>
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:33"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, </i>;<br/>


</div>
