<?xml version="1.0"?>
<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="E1:27"/><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>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b> non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  = <font color="Maroon">I</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">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</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">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E2:27"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:27_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
not <font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D18">PENCIL_1:def 18</a></i>;<br/>
<a NAME="E4:27_1"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> =  <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> <font color="Maroon">x</font></span>)</span> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p0</font> being   <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>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">p0</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="pzfmisc1.html#K1">}</a></span> as    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T11">PENCIL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,<font color="Maroon">C</font> as  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>by </b><i><a class="ref" href="pencil_1.html#T9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10">PENCIL_1:10</a>, <a class="ref" href="pencil_2.html#T7">PENCIL_2:7</a></i>;<br/>
<a NAME="E10:27_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:27_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E12:27_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E13:27_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</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">b</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">b</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:27_1"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:27_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>now </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="E1:27_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:27_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><b>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> <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_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="E1:27_1_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:27_1_1_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/><a NAME="E6:27_1_1_1_1_2"/>
not  <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> <font color="Maroon">i1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E7:27_1_1_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:27_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:27_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E8:27_1_1"/>
 <a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4">Segre_Blocks</a> <font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23">PENCIL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E9:27_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fB</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">b0</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="E18:27_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">fB</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b0</font> &amp; <font color="Maroon">b0</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">b0</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">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font>;<br/>

<b>let </b><font color="Maroon">B1</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<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="E19:27_1"/>
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/>

<b>hence </b><a NAME="E20:27_1"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:27_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:27_2"/><b>then </b>
<font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:27_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D18">PENCIL_1:def 18</a></i>;<br/>
<a NAME="E4:27_2"/><b>then </b>
not the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> =  <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> <font color="Maroon">x</font></span>)</span> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p0</font> being   <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>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">p0</font> as    <a href="pboole.html#M2">Element</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default"><font color="Maroon">p0</font></span><a href="pzfmisc1.html#K1">}</a></span> as    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T11">PENCIL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">x</font>,<font color="Maroon">C</font> as  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>by </b><i><a class="ref" href="pencil_1.html#T9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10">PENCIL_1:10</a>, <a class="ref" href="pencil_2.html#T7">PENCIL_2:7</a></i>;<br/>
<a NAME="E10:27_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:27_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E12:27_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E13:27_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">b</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">b</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">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:27_2"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:27_2_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:27_2_1"/><i><font color="Green">E106</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:27_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>now </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="E1:27_2_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:27_2_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a></i>;<br/><b>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> <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_1_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:27_2_1_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17">PBOOLE:def 17</a></i>;<br/><a NAME="E6:27_2_1_1_1_2"/>
not  <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> <font color="Maroon">i1</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><a NAME="E7:27_2_1_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:27_2_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:27_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E8:27_2_1"/>
 <a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4">Segre_Blocks</a> <font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23">PENCIL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E9:27_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</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>by </b><i><a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fB</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#T25">PENCIL_2:25</a></i>;<br/>
<b>consider </b><font color="Maroon">b0</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="E18:27_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">fB</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b0</font> &amp; <font color="Maroon">b0</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">b0</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">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font>;<br/>

<b>let </b><font color="Maroon">B1</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<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="E19:27_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E20:27_2"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E21:27_2"/><b>then </b>
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E22:27_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E23:27_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<b>hence </b><a NAME="E24:27_2"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">x'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x'</font>,<font color="Olive">y</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(), <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(), <font color="Maroon">x'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green">E141</font></i>: 
( ( for <font color="Olive">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</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">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( for <font color="Olive">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</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">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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="E3:27_3"/><i><font color="Green">E143</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fB1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">L1</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="E7:27_3"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">fB1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L1</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">L1</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">L1</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="E8:27_3"/><i><font color="Green">E145</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">b2</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="E10:27_3"/><i><font color="Green">E146</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</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> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><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><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">fB2</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</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>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">L2</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="E14:27_3"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">fB2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">L2</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">L2</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">L2</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="E15:27_3"/><i><font color="Green">E148</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  as   <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> <b>by </b><i><a class="ref" href="pencil_2.html#T13">PENCIL_2:13</a></i>;<br/>
<a NAME="E17:27_3"/><i><font color="Green">E149</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</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>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E18:27_3"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E19:27_3"/><b>then </b><i><font color="Green">E150</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a> &amp; <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E20:27_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E21:27_3"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, <a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E22:27_3"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">fB1</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/>
<a NAME="E23:27_3"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E24:27_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:27_3"/><b>then </b>
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E26:27_3"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E27:27_3"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">fB2</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/>
<a NAME="E28:27_3"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E29:27_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E30:27_3"/><b>then </b>
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E31:27_3"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">fB2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>thus </b><a NAME="E32:27_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , , , <a class="txt" href="pencil_3.html#E1:27"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:27"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:27"/><i><font color="Green">E160</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">y'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y'</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(), <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>(), <font color="Maroon">y'</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:27_4"/><i><font color="Green">E161</font></i>: 
( ( for <font color="Olive">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</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">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( for <font color="Olive">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</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">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</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="E3:27_4"/><i><font color="Green">E162</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fB1</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">L1</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="E7:27_4"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">fB1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">L1</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">L1</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">L1</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="E8:27_4"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:27_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:27"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">H<sub>1</sub></font>() st <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() holds <br/> ( <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <b>from </b><i><a class="ref" href="transgeo.html#S1">TRANSGEO:sch 1</a>(, , <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, );</i><br/>


</div>
