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<div class="add">

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



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b> non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c1">I</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">B1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2;<br/>
<a NAME="E2:27"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/>

<a NAME="E1:27_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:27_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
not <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D18" title="PENCIL_1:def.18">PENCIL_1:def 18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">p0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p0</font> = <font color="Maroon" title="c6">p0</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> = <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c7">p0</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> as    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T11" title="PENCIL_1:th.11">PENCIL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">C</font> as  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T9" title="PENCIL_1:th.9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10" title="PENCIL_1:th.10">PENCIL_1:10</a>, <a class="ref" href="pencil_2.html#T7" title="PENCIL_2:th.7">PENCIL_2:7</a></i>;<br/>
<a NAME="E9:27_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:27_1"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E11:27_1"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E12:27_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<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" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b</font> holds <br/><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:27_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:27_1_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:27_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E32">A7</font></i>: <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" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E2:27_1_1_1"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="pencil_3.html#E5:27_1_1"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</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" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c13">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:27_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="pencil_3.html#E2:27_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</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" title="c13">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:27_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:27_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:27_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a></i>;<br/><a NAME="E6:27_1_1_1_1_2"/>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<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" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E8:27_1_1_1_1_2"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:27_1_1_1_1_2"><i><font color="Green" title="E35">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:27_1_1"/>
 <a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4" title="PENCIL_1:func.4">Segre_Blocks</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23" title="PENCIL_1:def.23">PENCIL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="pencil_3.html#E5:27_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="pencil_3.html#E6:27_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">B</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E10:27_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="pencil_3.html#E11:27_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fB</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">B</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b0</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E16:27_1"/><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c11">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">b0</font> &amp; <font color="Maroon" title="c12">b0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c13">y</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font>;<br/>

<b>let </b><font color="Maroon" title="c14">B1</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>



<b>assume </b><a NAME="E17:27_1"/>
( <font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 ;<br/>

<b>hence </b><a NAME="E18:27_1"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c16">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="pencil_3.html#E11:27_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="pencil_3.html#E16:27_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="pencil_3.html#T24" target="_self" title="PENCIL_3:th.24">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:27"/><i><font color="Green" title="E28">A12</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">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" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/>

<a NAME="E1:27_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:27_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:27_2"/><b>then </b>
not <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D18" title="PENCIL_1:def.18">PENCIL_1:def 18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">C</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="struct_0.html#D9" title="STRUCT_0:def.9">STRUCT_0:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">p0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p0</font> = <font color="Maroon" title="c6">p0</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">Element</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_3.html#T6" target="_self" title="PENCIL_3:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> = <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c7">p0</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> as    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T11" title="PENCIL_1:th.11">PENCIL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">C</font> as  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="pencil_1.html#T9" title="PENCIL_1:th.9">PENCIL_1:9</a>, <a class="ref" href="pencil_1.html#T10" title="PENCIL_1:th.10">PENCIL_1:10</a>, <a class="ref" href="pencil_2.html#T7" title="PENCIL_2:th.7">PENCIL_2:7</a></i>;<br/>
<a NAME="E9:27_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:27_2"/><b>then </b><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">C</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E11:27_2"/><b>then </b><i><font color="Green" title="E30">A14</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E12:27_2"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<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" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_2_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:27_2_1"/><i><font color="Green" title="E31">A15</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b</font> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">b</font> holds <br/><font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:27_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:27_2_1"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:27_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E33">A17</font></i>: <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" title="c12">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:27_2_1_1"/><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><a NAME="E2:27_2_1_1"/><b>then </b><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="pencil_3.html#E5:27_2_1"><i><font color="Green" title="E32">A16</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">i1</font> = <font color="Maroon" title="c12">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27_2_1_1"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</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" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c13">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_2_1_1_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:27_2"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="pencil_3.html#E2:27_2_1_1"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</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" title="c13">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:27_2_1_1"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="pzfmisc1.html#D1" title="PZFMISC1:def.1">PZFMISC1:def 1</a></i>;<br/><a NAME="E4:27_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:27_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font>
 <b>by </b><i><a class="ref" href="pboole.html#D17" title="PBOOLE:def.17">PBOOLE:def 17</a></i>;<br/><a NAME="E6:27_2_1_1_1_2"/>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c13">i1</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<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" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E8:27_2_1_1_1_2"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:27_2_1_1_1_2"><i><font color="Green" title="E36">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:27_2_1"/>
 <a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4" title="PENCIL_1:func.4">Segre_Blocks</a> <font color="Maroon" title="c2">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23" title="PENCIL_1:def.23">PENCIL_1:def 23</a></i>;<br/>
<b>hence </b><a NAME="E8:27_2_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="pencil_3.html#E5:27_2_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="pencil_3.html#E6:27_2_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">B</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E10:27_2"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="pencil_3.html#E11:27_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">fB</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c10">B</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pencil_2.html#T25" title="PENCIL_2:th.25">PENCIL_2:25</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">b0</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E16:27_2"/><i><font color="Green" title="E31">A21</font></i>: 
( <font color="Maroon" title="c11">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">b0</font> &amp; <font color="Maroon" title="c12">b0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<b>take </b><font color="Maroon" title="c13">y</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font>;<br/>

<b>let </b><font color="Maroon" title="c14">B1</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c15">b1</font>, <font color="Maroon" title="c16">b2</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>



<b>assume </b><a NAME="E17:27_2"/><i><font color="Green" title="E32">A22</font></i>: 
( <font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c15">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c15">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> )
 ;<br/>

<a NAME="E18:27_2"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E19:27_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E20:27_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E21:27_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">fB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<b>hence </b><a NAME="E22:27_2"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c16">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="pencil_3.html#E11:27_2"><i><font color="Green" title="E30">A14</font></i></a>, <a class="txt" href="pencil_3.html#E16:27_2"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="pencil_3.html#E17:27_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="ref" href="pencil_3.html#T24" target="_self" title="PENCIL_3:th.24">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27"/><i><font color="Green" title="E29">A23</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">x'</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">x'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E30">A24</font></i>: 
( ( for <font color="Olive" title="b1">B1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> ) &amp; ( for <font color="Olive" title="b1">B1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">b1</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:27_3"/><i><font color="Green" title="E31">A25</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b1</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T8" target="_self" title="PENCIL_3:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B1</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b1</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_3"><i><font color="Green" title="E31">A25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">fB1</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">B1</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">L1</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E7:27_3"/><i><font color="Green" title="E32">A26</font></i>: 
( <font color="Maroon" title="c9">fB1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">L1</font> &amp; <font color="Maroon" title="c10">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E8:27_3"/><i><font color="Green" title="E33">A27</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:27_3"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="pencil_3.html#E3:27_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="pencil_3.html#E7:27_3"><i><font color="Green" title="E32">A26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">b2</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E10:27_3"/><i><font color="Green" title="E34">A28</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c11">b2</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T8" target="_self" title="PENCIL_3:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">B2</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c11">b2</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E10:27_3"><i><font color="Green" title="E34">A28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">fB2</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">B2</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">L2</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E14:27_3"/><i><font color="Green" title="E35">A29</font></i>: 
( <font color="Maroon" title="c13">fB2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c14">L2</font> &amp; <font color="Maroon" title="c14">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">L2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">L2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E15:27_3"/><i><font color="Green" title="E36">A30</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c14">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:27_3"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="pencil_3.html#E10:27_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="pencil_3.html#E14:27_3"><i><font color="Green" title="E35">A29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">ff</font> = <font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  as   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_2.html#T13" title="PENCIL_2:th.13">PENCIL_2:13</a></i>;<br/>
<a NAME="E17:27_3"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E18:27_3"/><b>then </b><i><font color="Green" title="E37">A31</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E19:27_3"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E20:27_3"/><b>then </b><i><font color="Green" title="E38">A32</font></i>: 
<font color="Maroon" title="c15">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="pencil_3.html#E18:27_3"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E21:27_3"/><b>then </b><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">fB1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">fB1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E18:27_3"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/>
<a NAME="E22:27_3"/><b>then </b><i><font color="Green" title="E40">A34</font></i>: 
<font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">fB1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">B1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E18:27_3"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/>
<a NAME="E23:27_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:27_3"/><b>then </b>
<font color="Maroon" title="c8">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">fB1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E21:27_3"><i><font color="Green" title="E39">A33</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E25:27_3"/><b>then </b><i><font color="Green" title="E41">A35</font></i>: 
<font color="Maroon" title="c8">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">fB1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E22:27_3"><i><font color="Green" title="E40">A34</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E26:27_3"/><i><font color="Green" title="E42">A36</font></i>: 
<font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">fB2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c13">fB2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E18:27_3"><i><font color="Green" title="E37">A31</font></i></a>, <a class="txt" href="pencil_3.html#E20:27_3"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/>
<a NAME="E27:27_3"/><b>then </b><i><font color="Green" title="E43">A37</font></i>: 
<font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">fB2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">B2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E18:27_3"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/>
<a NAME="E28:27_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E29:27_3"/><b>then </b>
<font color="Maroon" title="c12">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">fB2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E26:27_3"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E30:27_3"/><b>then </b>
<font color="Maroon" title="c12">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ff</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">fB2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E27:27_3"><i><font color="Green" title="E43">A37</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E31:27_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x'</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="pencil_3.html#E3:27_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="pencil_3.html#E7:27_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="pencil_3.html#E8:27_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="pencil_3.html#E10:27_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="pencil_3.html#E14:27_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="pencil_3.html#E15:27_3"><i><font color="Green" title="E36">A30</font></i></a>, <a class="txt" href="pencil_3.html#E25:27_3"><i><font color="Green" title="E41">A35</font></i></a>, <a class="ref" href="pencil_3.html#T24" target="_self" title="PENCIL_3:th.24">Th24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:27"/><i><font color="Green" title="E30">A38</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">y'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y'</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">y'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:27_4"/><i><font color="Green" title="E31">A39</font></i>: 
( ( for <font color="Olive" title="b1">B1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> ) &amp; ( for <font color="Olive" title="b1">B1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">b1</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b3">b2</font> &amp;  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b3">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">b1</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:27_4"/><i><font color="Green" title="E32">A40</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b1</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T8" target="_self" title="PENCIL_3:th.8">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B1</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b1</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:27_4"><i><font color="Green" title="E32">A40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">fB1</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">B1</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:27"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">L1</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E7:27_4"/><i><font color="Green" title="E33">A41</font></i>: 
( <font color="Maroon" title="c9">fB1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c10">L1</font> &amp; <font color="Maroon" title="c10">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E8:27_4"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c10">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:27_4"><i><font color="Green" title="E31">A39</font></i></a>, <a class="txt" href="pencil_3.html#E3:27_4"><i><font color="Green" title="E32">A40</font></i></a>, <a class="txt" href="pencil_3.html#E7:27_4"><i><font color="Green" title="E33">A41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:27_4"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:27_4"><i><font color="Green" title="E31">A39</font></i></a>, <a class="txt" href="pencil_3.html#E3:27_4"><i><font color="Green" title="E32">A40</font></i></a>, <a class="txt" href="pencil_3.html#E7:27_4"><i><font color="Green" title="E33">A41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:27"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon">H<sub>1</sub></font>() st <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">H<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] )
 <b>from </b><i><a class="ref" href="transgeo.html#S1" title="TRANSGEO:sch.1">TRANSGEO:sch 1</a>(<a class="txt" href="pencil_3.html#E2:27"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="pencil_3.html#E3:27"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="pencil_3.html#E4:27"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="pencil_3.html#E5:27"><i><font color="Green" title="E30">A38</font></i></a>);</i><br/>


</div>
