<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c5">s</font> =  <a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">i</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>;<br/>
<a NAME="E1:29_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
 <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="E2:29_1_1"/>
<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="E3:29_1_1"/><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="E4:29_1_1"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
 <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="E5:29_1_1"/>
<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="E6:29_1_1"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
<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="E7:29_1_1"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <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#E4:29_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">B1</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">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#E2:29"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B2</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#E1:29"><i><font color="Green" title="E27">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="c9">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="E11:29_1_1"/><i><font color="Green" title="E33">A7</font></i>: 
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">B2</font> &amp; <font color="Maroon" title="c9">b2</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="c9">b2</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="c9">b2</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="E12:29_1_1"/><i><font color="Green" title="E34">A8</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="pencil_3.html#D3" target="_self" title="PENCIL_3:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">j</font> =  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>;<br/>
<a NAME="E13:29_1_1"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c7">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">B2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a></i>;<br/>
<a NAME="E14:29_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">B1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/>
<a NAME="E15:29_1_1"/><b>then </b><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<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">J</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c4">b1</font></span>)</span></span>)</span>;<br/>
<a NAME="E16:29_1_1"/><i><font color="Green" title="E37">A11</font></i>: 
 <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/>
<a NAME="E17:29_1_1"/><i><font color="Green" title="E38">A12</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</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> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y1</font>, <font color="Maroon" title="c13">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green" title="E39">A13</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">y2</font>] )
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">bb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c14">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">bf</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:29_1_1_1"/><i><font color="Green" title="E41">A15</font></i>: 
( <font color="Maroon" title="c15">bf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">bf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> holds <br/><font color="Maroon" title="c15">bf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:29_1_1_1"/><i><font color="Green" title="E42">A16</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">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/>
<b>then reconsider </b><font color="Maroon" title="c16">bb</font> = <font color="Maroon" title="c14">bb</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">bbx</font> = <font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>;<br/>
<i><font color="Green" title="E43">A17</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> = 
<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 href="relat_1.html#K1" title="RELAT_1:func.1">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green" title="E44">A18</font></i>: 
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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/><b>then reconsider </b><font color="Maroon" title="c19">u</font> = <font color="Maroon" title="c18">o</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="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="29_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_2_1"/>
( <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c19">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_1"/><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1_2_1_1"><i><font color="Green" title="E45">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="29_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_2"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E45">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E4:29_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">b1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <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="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E5:29_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c18">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_2"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</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="c18">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_1_2_1_2"><i><font color="Green" title="E45">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c18">bbx1</font> = <font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1_1"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E11:29_1_1_1"/><i><font color="Green" title="E44">A21</font></i>: 
<font color="Maroon" title="c18">bbx1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<a NAME="E12:29_1_1_1"/><i><font color="Green" title="E45">A22</font></i>: 
<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="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_1_3"/>
<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="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">o</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="E1:29_1_1_1_3_1"/>
<font color="Maroon" title="c19">o</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="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c19">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E47">A24</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> = 
<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 href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_3_1_2"/>
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/><a NAME="E2:29_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E48">A25</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E3:29_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E49">A26</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">bb</font>
 <b>by </b><i><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="29_1_1_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_3_1_2_1"/>
( <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c20">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_1_3_1_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_1_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</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"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_3_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_3_1_2_1_1"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_3_1_2_1_2"/><i><font color="Green" title="E50">A28</font></i>: 
<font color="Maroon" title="c20">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_1_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_1"><i><font color="Green" title="E40">A14</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_1_3_1_2"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_3_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1_3_1_2_1_2"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1_1_1_3_1"/>
<font color="Maroon" title="c19">o</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1_1_3_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_1_3_1"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c19">o</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="E2:29_1_1_1_3"/>
<font color="Maroon" title="c19">o</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">u</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:29_1_1_1_3"/><i><font color="Green" title="E46">A29</font></i>: 
( <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">o</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</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/>
<i><font color="Green" title="E47">A30</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">u</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3"><i><font color="Green" title="E46">A29</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_3_3"/><i><font color="Green" title="E48">A31</font></i>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">u</font>
 ;<br/><a NAME="E2:29_1_1_1_3_3"/><b>then </b><i><font color="Green" title="E49">A32</font></i>: 
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3"><i><font color="Green" title="E46">A29</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">zz</font> = <font color="Maroon" title="c21">z</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#E4:29_1_1_1_3"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_3_3"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:29_1_1_1_3_3"/><i><font color="Green" title="E50">A33</font></i>: 
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_3_3"><i><font color="Green" title="E48">A31</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_3_3_1"/>
( <font color="Maroon" title="c22">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c22">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_3_3_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_3_3_1_1"/><i><font color="Green" title="E51">A34</font></i>: 
<font color="Maroon" title="c22">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</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"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_1_3_3"><i><font color="Green" title="E50">A33</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_1_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_3_3_1_1"/>
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_3_3_1_1"><i><font color="Green" title="E51">A34</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_3_3_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_3_3_1_2"/><i><font color="Green" title="E51">A35</font></i>: 
<font color="Maroon" title="c22">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_3_1_2"/><b>then </b>
( not <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">zz</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">zz</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_1_3_3_1_2"/><i><font color="Green" title="E52">A36</font></i>: 
<font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c23">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E5:29_1_1_1_3_3_1_2"/>
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3_3"><i><font color="Green" title="E50">A33</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_1_3_3_1_2"><i><font color="Green" title="E51">A35</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E6:29_1_1_1_3_3_1_2"/><b>then </b><i><font color="Green" title="E53">A37</font></i>: 
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3_3_1_2"><i><font color="Green" title="E52">A36</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:29_1_1_1_3_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1_3_3_1_2"><i><font color="Green" title="E51">A35</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E8:29_1_1_1_3_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</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"><font color="Maroon" title="c23">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_1"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_1"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_1_3_3"><i><font color="Green" title="E49">A32</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_1_3_3_1_2"><i><font color="Green" title="E52">A36</font></i></a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_1_3_3_1_2"/>
<font color="Maroon" title="c20">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_1_3_3_1_2"><i><font color="Green" title="E53">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:29_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c20">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_1_3"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1_1_1_3"/>
<font color="Maroon" title="c19">o</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="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_1_3"><i><font color="Green" title="E46">A29</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:29_1_1_1"/><i><font color="Green" title="E46">A38</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1_1"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">fbbx</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">bbx1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E15:29_1_1_1"/><i><font color="Green" title="E47">A39</font></i>: 
<font color="Maroon" title="c19">fbbx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1_1"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">y1</font> = 
<font color="Maroon" title="c19">fbbx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_1"><i><font color="Green" title="E45">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">y2</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_1"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="pencil_3.html#E15:29_1_1_1"><i><font color="Green" title="E47">A39</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_1"><i><font color="Green" title="E45">A22</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E18:29_1_1"/><i><font color="Green" title="E39">A40</font></i>: 
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> <font color="Maroon">H<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  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="29_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_2"/><i><font color="Green" title="E40">A41</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>()
 ;<br/>

<b>consider </b><font color="Maroon" title="c12">bb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green" title="E41">A42</font></i>: 
<font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">bf</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:29_1_1_2"/><i><font color="Green" title="E42">A43</font></i>: 
( <font color="Maroon" title="c13">bf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">bb</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">bf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> holds <br/><font color="Maroon" title="c13">bf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_2"><i><font color="Green" title="E41">A42</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:29_1_1_2"/><i><font color="Green" title="E43">A44</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">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/>
<b>then reconsider </b><font color="Maroon" title="c14">bb</font> = <font color="Maroon" title="c12">bb</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_2"><i><font color="Green" title="E42">A43</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">bbx</font> = <font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>;<br/>
<i><font color="Green" title="E44">A45</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> = 
<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 href="relat_1.html#K1" title="RELAT_1:func.1">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_2"/><i><font color="Green" title="E45">A46</font></i>: 
<font color="Maroon" title="c16">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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/><b>then reconsider </b><font color="Maroon" title="c17">u</font> = <font color="Maroon" title="c16">o</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="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="29_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_2_1"/>
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c17">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_1"/><i><font color="Green" title="E46">A47</font></i>: 
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_2"><i><font color="Green" title="E40">A41</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1_2"><i><font color="Green" title="E42">A43</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</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="c16">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_2_2_1_1"><i><font color="Green" title="E46">A47</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="29_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_2_1_2"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E46">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_2"><i><font color="Green" title="E41">A42</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><a NAME="E4:29_1_1_2_2_1_2"/>
<font color="Maroon" title="c4">b1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <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="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E5:29_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c16">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_2"><i><font color="Green" title="E45">A46</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</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="c16">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_2_2_1_2"><i><font color="Green" title="E46">A48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c16">bbx1</font> = <font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1_2"><i><font color="Green" title="E44">A45</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E11:29_1_1_2"/>
 <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="E12:29_1_1_2"/><b>then </b><i><font color="Green" title="E45">A49</font></i>: 
<font color="Maroon" title="c16">bbx1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
;<br/>
<a NAME="E13:29_1_1_2"/><i><font color="Green" title="E46">A50</font></i>: 
<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="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_2_3"/>
<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="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">o</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="E1:29_1_1_2_3_1"/>
<font color="Maroon" title="c17">o</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="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E47">A51</font></i>: 
<font color="Maroon" title="c17">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E48">A52</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> = 
<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 href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_3_1_2"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/><a NAME="E2:29_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E49">A53</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E3:29_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E50">A54</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">bb</font>
 <b>by </b><i><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="29_1_1_2_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_3_1_2_1"/>
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c18">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_3_1_2_1_1"/><i><font color="Green" title="E51">A55</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_2_3_1_2"><i><font color="Green" title="E50">A54</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_2_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</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"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_3_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_3_1_2_1_1"><i><font color="Green" title="E51">A55</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_3_1_2_1_2"/><i><font color="Green" title="E51">A56</font></i>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_2_3_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_2"><i><font color="Green" title="E41">A42</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_2_3_1_2"><i><font color="Green" title="E49">A53</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_3_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_2_3_1_2_1_2"><i><font color="Green" title="E51">A56</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1_1_2_3_1"/>
<font color="Maroon" title="c17">o</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1_2_3_1"><i><font color="Green" title="E47">A51</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_2_3_1"><i><font color="Green" title="E48">A52</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c17">o</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="E2:29_1_1_2_3"/>
<font color="Maroon" title="c17">o</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">u</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:29_1_1_2_3"/><i><font color="Green" title="E47">A57</font></i>: 
( <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">o</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; ( for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</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/>
<i><font color="Green" title="E48">A58</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">u</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3"><i><font color="Green" title="E47">A57</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_3_3"/><i><font color="Green" title="E49">A59</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">u</font>
 ;<br/><a NAME="E2:29_1_1_2_3_3"/><b>then </b><i><font color="Green" title="E50">A60</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3"><i><font color="Green" title="E47">A57</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">zz</font> = <font color="Maroon" title="c19">z</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#E4:29_1_1_2_3"><i><font color="Green" title="E47">A57</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_3_3"><i><font color="Green" title="E49">A59</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:29_1_1_2_3_3"/><i><font color="Green" title="E51">A61</font></i>: 
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3"><i><font color="Green" title="E47">A57</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_3_3"><i><font color="Green" title="E49">A59</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_2_3_3_1"/>
( <font color="Maroon" title="c20">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c20">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_3_3_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_2_3_3_1_1"/><i><font color="Green" title="E52">A62</font></i>: 
<font color="Maroon" title="c20">zz</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</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"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_2_3_3"><i><font color="Green" title="E51">A61</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_2_3_3_1_1"/><b>then </b>
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_3_3_1_1"/>
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_2"><i><font color="Green" title="E42">A43</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_3_3_1_1"><i><font color="Green" title="E52">A62</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2_3_3_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2_3_3_1_2"/><i><font color="Green" title="E52">A63</font></i>: 
<font color="Maroon" title="c20">zz</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_3_1_2"/><b>then </b>
( not <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">zz</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">zz</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_2_3_3_1_2"/><i><font color="Green" title="E53">A64</font></i>: 
<font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/><a NAME="E5:29_1_1_2_3_3_1_2"/>
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3_3"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_2_3_3_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E6:29_1_1_2_3_3_1_2"/><b>then </b><i><font color="Green" title="E54">A65</font></i>: 
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3_3_1_2"><i><font color="Green" title="E53">A64</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:29_1_1_2_3_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_2_3_3_1_2"><i><font color="Green" title="E52">A63</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E8:29_1_1_2_3_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</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"><font color="Maroon" title="c21">o</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_2"><i><font color="Green" title="E42">A43</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_2"><i><font color="Green" title="E43">A44</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_2_3_3"><i><font color="Green" title="E50">A60</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_2_3_3_1_2"><i><font color="Green" title="E53">A64</font></i></a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_2_3_3_1_2"/>
<font color="Maroon" title="c18">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_2_3_3_1_2"><i><font color="Green" title="E54">A65</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:29_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c18">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1_2_3"><i><font color="Green" title="E48">A58</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1_1_2_3"/>
<font color="Maroon" title="c17">o</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="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_2_3"><i><font color="Green" title="E47">A57</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29_1_1_2"/><i><font color="Green" title="E47">A66</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">bb</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c11">x</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1_2"><i><font color="Green" title="E45">A49</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">fbbx</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">bbx1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<b>take </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">fbbx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>;<br/>

<b>let </b><font color="Maroon" title="c19">J</font> be    <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="E16:29_1_1_2"/><i><font color="Green" title="E48">A67</font></i>: 
<font color="Maroon" title="c19">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E17:29_1_1_2"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_2"><i><font color="Green" title="E47">A66</font></i></a>, <a class="txt" href="pencil_3.html#E16:29_1_1_2"><i><font color="Green" title="E48">A67</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E13:29_1_1_2"><i><font color="Green" title="E46">A50</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">M</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E20:29_1_1"/><i><font color="Green" title="E40">A68</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></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> <font color="Maroon">H<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="pencil_3.html#E17:29_1_1"><i><font color="Green" title="E38">A12</font></i></a>, <a class="txt" href="pencil_3.html#E18:29_1_1"><i><font color="Green" title="E39">A40</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c12">t</font> =  <a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font>;<br/>
<a NAME="E21:29_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">M</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</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="E1:29_1_1_3"/>
<font color="Maroon" title="c13">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="c11">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_3"/><i><font color="Green" title="E41">A69</font></i>: 
( <font color="Maroon" title="c14">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">M</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">o</font> = <font color="Maroon" title="c14">o</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_3"><i><font color="Green" title="E41">A69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">p</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E6:29_1_1_3"/><i><font color="Green" title="E42">A70</font></i>: 
( <font color="Maroon" title="c16">p</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="c4">b1</font> &amp; <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="c16">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c15">o</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">pio</font> = <font color="Maroon" title="c16">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c15">o</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_3"><i><font color="Green" title="E42">A70</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<a NAME="E8:29_1_1_3"/><i><font color="Green" title="E43">A71</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">p</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c15">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pio</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">J</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">pio</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E10:29_1_1_3"/><i><font color="Green" title="E44">A72</font></i>: 
<font color="Maroon" title="c18">J</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1_3"><i><font color="Green" title="E43">A71</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E6:29_1_1_3"><i><font color="Green" title="E42">A70</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_3"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 
;<br/>
<a NAME="E12:29_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E13:29_1_1_3"/><b>then </b>
<font color="Maroon" title="c18">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E14:29_1_1_3"/><b>then </b>
<font color="Maroon" title="c18">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 
<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="E15:29_1_1_3"/>
<font color="Maroon" title="c13">x</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_3"><i><font color="Green" title="E41">A69</font></i></a>, <a class="txt" href="pencil_3.html#E10:29_1_1_3"><i><font color="Green" title="E44">A72</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">M</font> = <font color="Maroon" title="c11">M</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <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="c4">b1</font></span>)</span></span>)</span>,<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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">M</font>
;<br/>

<b>set </b><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c13">M</font>;<br/>
<b>thus </b><a NAME="E23:29_1_1"/>
<font color="Maroon" title="c13">M</font> is <a href="pencil_2.html#V1" title="PENCIL_2:attr.1">isomorphic</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_1_4"/><i><font color="Green" title="E41">A73</font></i>: 
<font color="Maroon" title="c13">M</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c16">x2</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_1"/><i><font color="Green" title="E42">A74</font></i>: 
( <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">M</font> &amp; <font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">M</font> &amp; <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c17">o1</font> = <font color="Maroon" title="c15">x1</font>, <font color="Maroon" title="c18">o2</font> = <font color="Maroon" title="c16">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c19">p1</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E4:29_1_1_4_1"/><i><font color="Green" title="E43">A75</font></i>: 
( <font color="Maroon" title="c19">p1</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="c4">b1</font> &amp; <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="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">o1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">p1io</font> = <font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_4_1"><i><font color="Green" title="E43">A75</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<a NAME="E6:29_1_1_4_1"/><i><font color="Green" title="E44">A76</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">p1io</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">J1</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">p1io</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_4_1"/><i><font color="Green" title="E45">A77</font></i>: 
<font color="Maroon" title="c21">J1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">o1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_1"><i><font color="Green" title="E44">A76</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E4:29_1_1_4_1"><i><font color="Green" title="E43">A75</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">p2</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E10:29_1_1_4_1"/><i><font color="Green" title="E46">A78</font></i>: 
( <font color="Maroon" title="c22">p2</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="c4">b1</font> &amp; <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="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">o2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">p2io</font> = <font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E10:29_1_1_4_1"><i><font color="Green" title="E46">A78</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<a NAME="E12:29_1_1_4_1"/><i><font color="Green" title="E47">A79</font></i>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p2io</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">J2</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p2io</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E14:29_1_1_4_1"/>
<font color="Maroon" title="c24">J2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">o2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1_4_1"><i><font color="Green" title="E47">A79</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E10:29_1_1_4_1"><i><font color="Green" title="E46">A78</font></i></a></i>;<br/>
<a NAME="E15:29_1_1_4_1"/><b>then </b><i><font color="Green" title="E48">A80</font></i>: 
<font color="Maroon" title="c13">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">J2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a></i>;<br/>
<a NAME="E16:29_1_1_4_1"/><i><font color="Green" title="E49">A81</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_1"/><i><font color="Green" title="E50">A82</font></i>: 
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_1_1_1"/>
( <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_1_1_1"/><i><font color="Green" title="E51">A83</font></i>: 
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">o1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1_4_1"><i><font color="Green" title="E49">A81</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_1_1_1_1"><i><font color="Green" title="E51">A83</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_1_1_2"/>
<font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_4_1"><i><font color="Green" title="E43">A75</font></i></a>, <a class="txt" href="pencil_3.html#E16:29_1_1_4_1"><i><font color="Green" title="E49">A81</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_1_1"><i><font color="Green" title="E50">A82</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E18:29_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c20">p1io</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="c4">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1_4_1"><i><font color="Green" title="E49">A81</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E19:29_1_1_4_1"/><b>then </b><i><font color="Green" title="E50">A84</font></i>: 
<font color="Maroon" title="c21">J1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E20:29_1_1_4_1"/><i><font color="Green" title="E51">A85</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_2"/><i><font color="Green" title="E52">A86</font></i>: 
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_1_2_1"/>
( <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_2_1_1"/><i><font color="Green" title="E53">A87</font></i>: 
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1_4_1"><i><font color="Green" title="E51">A85</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_1_2_1_1"><i><font color="Green" title="E53">A87</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:29_1_1_4_1"><i><font color="Green" title="E46">A78</font></i></a>, <a class="txt" href="pencil_3.html#E20:29_1_1_4_1"><i><font color="Green" title="E51">A85</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_1_2"><i><font color="Green" title="E52">A86</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:29_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c23">p2io</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="c4">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1_4_1"><i><font color="Green" title="E51">A85</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E23:29_1_1_4_1"/><b>then </b><i><font color="Green" title="E52">A88</font></i>: 
<font color="Maroon" title="c24">J2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E24:29_1_1_4_1"/><i><font color="Green" title="E53">A89</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c21">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">J2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_3"/>
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c26">l</font> = <font color="Maroon" title="c25">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_1_3_1"/>
( <font color="Maroon" title="c26">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c26">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_3_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_3_1_1"/>
<font color="Maroon" title="c26">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_1_3_1_1"/>
<font color="Maroon" title="c21">J1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">J2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_1"><i><font color="Green" title="E42">A74</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1_4_1"><i><font color="Green" title="E45">A77</font></i></a>, <a class="txt" href="pencil_3.html#E15:29_1_1_4_1"><i><font color="Green" title="E48">A80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_1_3_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_3_1_2"/>
<font color="Maroon" title="c26">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_1_3_1_2"/>
<font color="Maroon" title="c21">J1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">J2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="pencil_3.html#E19:29_1_1_4_1"><i><font color="Green" title="E50">A84</font></i></a>, <a class="txt" href="pencil_3.html#E23:29_1_1_4_1"><i><font color="Green" title="E52">A88</font></i></a>, <a class="ref" href="pencil_1.html#T23" title="PENCIL_1:th.23">PENCIL_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E26:29_1_1_4_1"/><b>then </b><i><font color="Green" title="E54">A90</font></i>: 
<font color="Maroon" title="c21">J1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">J2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E24:29_1_1_4_1"><i><font color="Green" title="E53">A89</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E27:29_1_1_4_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">p2</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="E28:29_1_1_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">p1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c17">o1</font></span>)</span> <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="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">o1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c22">p2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c18">o2</font></span>)</span> <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="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o2</font> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E29:29_1_1_4_1"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="pencil_3.html#E26:29_1_1_4_1"><i><font color="Green" title="E54">A90</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29_1_1_4"/><i><font color="Green" title="E42">A91</font></i>: 
<font color="Maroon" title="c13">M</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_4_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">M</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_4_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">M</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:29_1_1_4_2_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</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="E1:29_1_1_4_2_1_1"/>
<font color="Maroon" title="c15">x</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c15">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c17">p0</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E4:29_1_1_4_2_1_1"/><i><font color="Green" title="E43">A92</font></i>: 
( <font color="Maroon" title="c17">p0</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">b2</font> &amp; <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="c17">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">x1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c16">x1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">p</font> = <font color="Maroon" title="c17">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">x1</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_4_2_1_1"><i><font color="Green" title="E43">A92</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">p1</font> = <font color="Maroon" title="c18">p</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<a NAME="E7:29_1_1_4_2_1_1"/><i><font color="Green" title="E44">A93</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E9:29_1_1_4_2_1_1"/><i><font color="Green" title="E45">A94</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">M</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"><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="c4">b1</font></span>)</span></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="E10:29_1_1_4_2_1_1"/>
<font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c20">q</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="c4">b1</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E12:29_1_1_4_2_1_1"/><b>then </b><i><font color="Green" title="E46">A95</font></i>: 
<font color="Maroon" title="c20">q</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="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<a NAME="E13:29_1_1_4_2_1_1"/><i><font color="Green" title="E47">A96</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">p0</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="E14:29_1_1_4_2_1_1"/><i><font color="Green" title="E48">A97</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_2_1_1_1"/>
<font color="Maroon" title="c21">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c22">l</font> = <font color="Maroon" title="c21">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_2_1_1_1_1"/>
( <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c22">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_2_1_1_1_1_1"/>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_2_1_1"><i><font color="Green" title="E48">A97</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_2_1_1_1_1_1"/>
<font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_2_1_1_1_1_2"/>
<font color="Maroon" title="c22">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">l</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_4_2_1_1_1_1_2"/><i><font color="Green" title="E50">A99</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">p0</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c23">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_2_1_1_1_1_2_1"/><i><font color="Green" title="E51">A100</font></i>: 
<font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_2_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c23">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E52">A101</font></i>: 
<font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E50">A99</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E52">A101</font></i></a>, <a class="ref" href="pencil_3.html#D3" target="_self" title="PENCIL_3:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c23">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_2_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c19">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_4_2_1_1"><i><font color="Green" title="E43">A92</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E50">A99</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_2_1_1_1_1_2_1"><i><font color="Green" title="E51">A100</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:29_1_1_4_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">p</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">b2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E50">A99</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">q0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:29_1_1_4_2_1_1_1_1_2"/><i><font color="Green" title="E51">A102</font></i>: 
( <font color="Maroon" title="c23">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c23">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">B1</font> &amp; <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">q0</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:29_1_1_4_2_1_1_1_1_2"/>
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">q0</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1_4_2_1_1"><i><font color="Green" title="E44">A93</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E51">A102</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_4_2_1_1_1_1_2"/>
<font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_2_1_1"><i><font color="Green" title="E48">A97</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E49">A98</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1_4_2_1_1_1_1_2"><i><font color="Green" title="E51">A102</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E16:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c20">q</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_2_1_1"><i><font color="Green" title="E48">A97</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E17:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c17">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1_4_2_1_1"><i><font color="Green" title="E44">A93</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E18:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">x1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_4_2_1_1"><i><font color="Green" title="E46">A95</font></i></a></i>;<br/>
<a NAME="E19:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_2_1_1"><i><font color="Green" title="E47">A96</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E20:29_1_1_4_2_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">M</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:29_1_1_4_2_1_1"><i><font color="Green" title="E45">A94</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_4_2_1_1"><i><font color="Green" title="E46">A95</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


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


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1_4"/><i><font color="Green" title="E43">A103</font></i>: 
<font color="Maroon" title="c13">M</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_4"><i><font color="Green" title="E41">A73</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a><br/>

<a NAME="E4:29_1_1_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">M</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"><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="c9">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4"><i><font color="Green" title="E42">A91</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:29_1_1_4"/><b>then </b><i><font color="Green" title="E44">A104</font></i>: 
<font color="Maroon" title="c13">M</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4"><i><font color="Green" title="E41">A73</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<b>thus </b><a NAME="E6:29_1_1_4"/>
<font color="Maroon" title="c13">M</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">S0</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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="c4">b1</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_3"/>
<font color="Maroon" title="c15">S0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">S</font> = <font color="Maroon" title="c15">S0</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <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="c4">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">L</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</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_3.html#T12" target="_self" title="PENCIL_3:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">K</font> = <font color="Maroon" title="c3">f</font> <a href="pencil_2.html#K2" title="PENCIL_2:func.2">.:</a> <font color="Maroon" title="c17">L</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</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>consider </b><font color="Maroon" title="c19">k</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="E6:29_1_1_4_3"/><i><font color="Green" title="E45">A105</font></i>: 
( <font color="Maroon" title="c18">K</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="c19">k</font> &amp; <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <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="c19">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24" title="PENCIL_1:th.24">PENCIL_1:24</a></i>;<br/>
<a NAME="E7:29_1_1_4_3"/><i><font color="Green" title="E46">A106</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">k</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="E8:29_1_1_4_3"/><i><font color="Green" title="E47">A107</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_3_1_1"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_1_1_1"/><i><font color="Green" title="E48">A108</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1_4_3"><i><font color="Green" title="E47">A107</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">b1</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_3_1_1_1"><i><font color="Green" title="E48">A108</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_1_1_2"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:29_1_1_4_3"/><b>then </b><i><font color="Green" title="E48">A109</font></i>: 
<font color="Maroon" title="c17">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1_4_3"><i><font color="Green" title="E47">A107</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>;<br/>
<a NAME="E11:29_1_1_4_3"/><b>then </b>
<font color="Maroon" title="c18">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E12:29_1_1_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">K</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:29_1_1_4_3"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E14:29_1_1_4_3"/><b>then </b><i><font color="Green" title="E49">A110</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> holds <br/><font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="pencil_1.html#T26" title="PENCIL_1:th.26">PENCIL_1:26</a></i>;<br/>
<a NAME="E15:29_1_1_4_3"/>
<font color="Maroon" title="c13">M</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c16">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_4_3_2"/>
<font color="Maroon" title="c13">M</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c16">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">o</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="E1:29_1_1_4_3_2_1"/>
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c16">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_4_3_2_1"/><i><font color="Green" title="E50">A111</font></i>: 
( <font color="Maroon" title="c21">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">M</font> &amp; <font color="Maroon" title="c21">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font> &amp; <font color="Maroon" title="c20">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">u</font> = <font color="Maroon" title="c21">u</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_3_2_1"><i><font color="Green" title="E50">A111</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">p0</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E6:29_1_1_4_3_2_1"/><i><font color="Green" title="E51">A112</font></i>: 
( <font color="Maroon" title="c23">p0</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="c4">b1</font> &amp; <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="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_3_2_1"><i><font color="Green" title="E51">A112</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E9:29_1_1_4_3_2_1"/><i><font color="Green" title="E52">A113</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c23">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_1_1"/><i><font color="Green" title="E53">A114</font></i>: 
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_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="E1:29_1_1_4_3_2_1_1_1"/>
( <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c26">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_1_1_1_1"/><i><font color="Green" title="E54">A115</font></i>: 
<font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:29_1_1_4_3_2_1"><i><font color="Green" title="E52">A113</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_3_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_3_2_1"><i><font color="Green" title="E50">A111</font></i></a>, <a class="txt" href="pencil_3.html#E9:29_1_1_4_3_2_1"><i><font color="Green" title="E52">A113</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_3_2_1_1_1_1"><i><font color="Green" title="E54">A115</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_1_1_1_2"/><i><font color="Green" title="E54">A116</font></i>: 
<font color="Maroon" title="c26">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_4_3_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3_2_1"><i><font color="Green" title="E51">A112</font></i></a>, <a class="txt" href="pencil_3.html#E9:29_1_1_4_3_2_1"><i><font color="Green" title="E52">A113</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_3_2_1_1"><i><font color="Green" title="E53">A114</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_3_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_4_3_2_1_1_1_2"><i><font color="Green" title="E54">A116</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E11:29_1_1_4_3_2_1"/><b>then </b>
<font color="Maroon" title="c24">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E9:29_1_1_4_3_2_1"><i><font color="Green" title="E52">A113</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E12:29_1_1_4_3_2_1"/><b>then </b><i><font color="Green" title="E53">A117</font></i>: 
<font color="Maroon" title="c25">q</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="c19">k</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E13:29_1_1_4_3_2_1"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c22">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<a NAME="E14:29_1_1_4_3_2_1"/><b>then </b>
<font color="Maroon" title="c25">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c22">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_3_2_1"><i><font color="Green" title="E51">A112</font></i></a></i>;<br/>
<a NAME="E15:29_1_1_4_3_2_1"/><b>then </b>
<font color="Maroon" title="c13">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:29_1_1_4_3_2_1"/>
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1_4_3"><i><font color="Green" title="E46">A106</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1_4_3"><i><font color="Green" title="E49">A110</font></i></a>, <a class="txt" href="pencil_3.html#E3:29_1_1_4_3_2_1"><i><font color="Green" title="E50">A111</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_4_3_2_1"><i><font color="Green" title="E53">A117</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c20">o</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="E2:29_1_1_4_3_2"/><i><font color="Green" title="E50">A118</font></i>: 
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 ;<br/>

<a NAME="E3:29_1_1_4_3_2"/>
<font color="Maroon" title="c19">k</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="c19">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1_4_3"><i><font color="Green" title="E49">A110</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c20">o</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1_4_3_2"><i><font color="Green" title="E50">A118</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">p0</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E6:29_1_1_4_3_2"/><i><font color="Green" title="E51">A119</font></i>: 
( <font color="Maroon" title="c22">p0</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="c19">k</font> &amp; <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="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">k</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_3"><i><font color="Green" title="E49">A110</font></i></a>, <a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<a NAME="E7:29_1_1_4_3_2"/>
<font color="Maroon" title="c18">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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>then reconsider </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_3_2"><i><font color="Green" title="E51">A119</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E10:29_1_1_4_3_2"/><i><font color="Green" title="E52">A120</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">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> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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="E11:29_1_1_4_3_2"/><i><font color="Green" title="E53">A121</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">p0</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_2"/><i><font color="Green" title="E54">A122</font></i>: 
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_3_2_2_1"/>
( <font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span> or <font color="Maroon" title="c25">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_2_1_1"/>
<font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_3_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_3"><i><font color="Green" title="E49">A110</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4_3_2"><i><font color="Green" title="E50">A118</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1_4_3_2"><i><font color="Green" title="E53">A121</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_2_1_2"/>
<font color="Maroon" title="c25">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_3_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3_2"><i><font color="Green" title="E51">A119</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1_4_3_2"><i><font color="Green" title="E53">A121</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_3_2_2"><i><font color="Green" title="E54">A122</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="pencil_2.html#K2" title="PENCIL_2:func.2">.:</a> <font color="Maroon" title="c17">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1_4_3_2"><i><font color="Green" title="E53">A121</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c25">qq</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:29_1_1_4_3_2"/><i><font color="Green" title="E54">A123</font></i>: 
( <font color="Maroon" title="c25">qq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c25">qq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">L</font> &amp; <font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">qq</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E16:29_1_1_4_3_2"/><i><font color="Green" title="E55">A124</font></i>: 
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">q</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E17:29_1_1_4_3_2"/><b>then </b><i><font color="Green" title="E56">A125</font></i>: 
<font color="Maroon" title="c24">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="pencil_3.html#E15:29_1_1_4_3_2"><i><font color="Green" title="E54">A123</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E18:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c24">q</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="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <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="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E10:29_1_1_4_3_2"><i><font color="Green" title="E52">A120</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E19:29_1_1_4_3_2"/><b>then </b><i><font color="Green" title="E57">A126</font></i>: 
<font color="Maroon" title="c24">q</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="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1_4_3"><i><font color="Green" title="E47">A107</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">qi</font> = <font color="Maroon" title="c24">q</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="c4">b1</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c27">q0</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E22:29_1_1_4_3_2"/><i><font color="Green" title="E58">A127</font></i>: 
( <font color="Maroon" title="c27">q0</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="c4">b1</font> &amp; <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="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">qi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<a NAME="E23:29_1_1_4_3_2"/><i><font color="Green" title="E59">A128</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">q0</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_3_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c28">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_3"/>
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_3_2_3_1"/>
( <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c28">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_3_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_3_1_1"/>
<font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_3_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E23:29_1_1_4_3_2"><i><font color="Green" title="E59">A128</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_3_2_3_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_3_2_3_1_2"/><i><font color="Green" title="E60">A129</font></i>: 
<font color="Maroon" title="c28">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">q0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_3_2_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:29_1_1_4_3"><i><font color="Green" title="E48">A109</font></i></a>, <a class="txt" href="pencil_3.html#E17:29_1_1_4_3_2"><i><font color="Green" title="E56">A125</font></i></a>, <a class="txt" href="pencil_3.html#E22:29_1_1_4_3_2"><i><font color="Green" title="E58">A127</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_3_2_3_1_2"><i><font color="Green" title="E60">A129</font></i></a>, <a class="ref" href="pencil_1.html#T23" title="PENCIL_1:th.23">PENCIL_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c24">q</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E23:29_1_1_4_3_2"><i><font color="Green" title="E59">A128</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E26:29_1_1_4_3_2"/><b>then </b>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c27">q0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c26">qi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">q</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<a NAME="E27:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c26">qi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1_4_3_2"><i><font color="Green" title="E55">A124</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E22:29_1_1_4_3_2"><i><font color="Green" title="E58">A127</font></i></a></i>;<br/>
<a NAME="E28:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a></i>;<br/>
<a NAME="E29:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">o</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1_4_3_2"><i><font color="Green" title="E53">A121</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E30:29_1_1_4_3_2"/>
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c16">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E19:29_1_1_4_3_2"><i><font color="Green" title="E57">A126</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:29_1_1_4_3"/>
<font color="Maroon" title="c13">M</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">S0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_3"><i><font color="Green" title="E45">A105</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1_4_3"><i><font color="Green" title="E49">A110</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E7:29_1_1_4"/>
<font color="Maroon" title="c13">M</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4"><i><font color="Green" title="E43">A103</font></i></a>, <a class="ref" href="pencil_2.html#T12" title="PENCIL_2:th.12">PENCIL_2:12</a></i>;<br/>

<b>thus </b><a NAME="E8:29_1_1_4"/>
<font color="Maroon" title="c13">M</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">S0</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_4"/>
<font color="Maroon" title="c15">S0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">S</font> = <font color="Maroon" title="c15">S0</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">L</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</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#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="pencil_3.html#T12" target="_self" title="PENCIL_3:th.12">Th12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">K</font> = <font color="Maroon" title="c3">f</font> <a href="pencil_2.html#K3" title="PENCIL_2:func.3">"</a> <font color="Maroon" title="c17">L</font> as   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</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>consider </b><font color="Maroon" title="c19">k</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="E6:29_1_1_4_4"/><i><font color="Green" title="E45">A130</font></i>: 
( <font color="Maroon" title="c18">K</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="c19">k</font> &amp; <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <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="c19">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24" title="PENCIL_1:th.24">PENCIL_1:24</a></i>;<br/>
<a NAME="E7:29_1_1_4_4"/><i><font color="Green" title="E46">A131</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_4_1_1"/>
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_1_1_1"/><i><font color="Green" title="E47">A132</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1_4_4"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</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"><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"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_4_1_1_1"><i><font color="Green" title="E47">A132</font></i></a>, <a class="ref" href="pencil_3.html#D3" target="_self" title="PENCIL_3:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_1_1_2"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E9:29_1_1_4_4"/><b>then </b>
<font color="Maroon" title="c17">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1_4_4"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></i>;<br/>
<a NAME="E10:29_1_1_4_4"/><b>then </b>
<font color="Maroon" title="c18">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="pencil_3.html#E15:29_1_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<a NAME="E11:29_1_1_4_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">K</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_4"><i><font color="Green" title="E45">A130</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E12:29_1_1_4_4"/><b>then </b><i><font color="Green" title="E47">A133</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6" title="PENCIL_1:def.6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E13:29_1_1_4_4"/><b>then </b><i><font color="Green" title="E48">A134</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> holds <br/><font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="ref" href="pencil_1.html#T26" title="PENCIL_1:th.26">PENCIL_1:26</a></i>;<br/>
<a NAME="E14:29_1_1_4_4"/>
<font color="Maroon" title="c13">M</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1_1_4_4_2"/>
<font color="Maroon" title="c13">M</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">o</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="E1:29_1_1_4_4_2_1"/><i><font color="Green" title="E49">A135</font></i>: 
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">S</font>
 ;<br/>

<a NAME="E2:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green" title="E50">A136</font></i>: 
( <font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">M</font> &amp; <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c20">o</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_4_4_2_1"><i><font color="Green" title="E49">A135</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">p</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E5:29_1_1_4_4_2_1"/><i><font color="Green" title="E51">A137</font></i>: 
( <font color="Maroon" title="c22">p</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="c4">b1</font> &amp; <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="c22">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">q</font> = <font color="Maroon" title="c22">p</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1_4_4_2_1"><i><font color="Green" title="E51">A137</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">fq</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">q</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_4_4_2_1"/><i><font color="Green" title="E52">A138</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">k</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="E9:29_1_1_4_4_2_1"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">q</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<a NAME="E10:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green" title="E53">A139</font></i>: 
<font color="Maroon" title="c24">fq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E5:29_1_1_4_4_2_1"><i><font color="Green" title="E51">A137</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green" title="E54">A140</font></i>: 
<font color="Maroon" title="c24">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4_4_2_1"><i><font color="Green" title="E50">A136</font></i></a></i>;<br/>
<a NAME="E12:29_1_1_4_4_2_1"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">b1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c25">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="E1:29_1_1_4_4_2_1_1"/>
<font color="Maroon" title="c25">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c26">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:29_1_1_4_4_2_1_1"/><i><font color="Green" title="E55">A141</font></i>: 
( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</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="E4:29_1_1_4_4_2_1_1"/><i><font color="Green" title="E56">A142</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></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="txt" href="pencil_3.html#E3:29_1_1_4_4_2_1_1"><i><font color="Green" title="E55">A141</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="hs2(this)" href="javascript:()" title="29_1_1_4_4_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_1_1_1"/>
<font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:29_1_1_4_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E57">A143</font></i>: 
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_4_2_1_1"><i><font color="Green" title="E55">A141</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_4_4_2_1_1"><i><font color="Green" title="E56">A142</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_4_2_1_1_1_1"/>
( <font color="Maroon" title="c27">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c27">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_1_1_1_1_1"/><i><font color="Green" title="E58">A144</font></i>: 
<font color="Maroon" title="c27">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:29_1_1_4_4_2_1_1"><i><font color="Green" title="E56">A142</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4_4_2_1_1_1"><i><font color="Green" title="E57">A143</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_4_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:29_1_1_4_4_2_1_1_1_1_1"/>
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_4_2_1_1_1_1_1"><i><font color="Green" title="E58">A144</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_1_1_1_1_2"/>
<font color="Maroon" title="c27">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_4_2_1_1_1_1_2"/>
<font color="Maroon" title="c26">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1_4_4_2_1_1_1"><i><font color="Green" title="E57">A143</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:29_1_1_4_4_2_1_1"/>
<font color="Maroon" title="c25">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="c4">b1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:29_1_1_4_4_2_1_1"><i><font color="Green" title="E55">A141</font></i></a>, <a class="txt" href="pencil_3.html#E4:29_1_1_4_4_2_1_1"><i><font color="Green" title="E56">A142</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green" title="E55">A145</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E11:29_1_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E14:29_1_1_4_4_2_1"/><i><font color="Green" title="E56">A146</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">fq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b2</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_1_2"/><i><font color="Green" title="E57">A147</font></i>: 
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_4_2_1_2_1"/>
( <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> or <font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_1_2_1_1"/>
<font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_4_2_1_2_1_1"/>
<font color="Maroon" title="c24">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1_4_4_2_1"><i><font color="Green" title="E54">A140</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1_4_4_2_1"><i><font color="Green" title="E56">A146</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_1_2_1_2"/>
<font color="Maroon" title="c25">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b2</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_4_2_1_2_1_2"/>
<font color="Maroon" title="c24">fq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:29_1_1_4_4_2_1"><i><font color="Green" title="E53">A139</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4_2_1"><i><font color="Green" title="E55">A145</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1_4_4_2_1"><i><font color="Green" title="E56">A146</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_4_2_1_2"><i><font color="Green" title="E57">A147</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E16:29_1_1_4_4_2_1"/><b>then </b>
<font color="Maroon" title="c24">fq</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1_4_4_2_1"><i><font color="Green" title="E56">A146</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E17:29_1_1_4_4_2_1"/><b>then </b>
<font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">K</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E18:29_1_1_4_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <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="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</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="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_4"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1_4_4_2_1"><i><font color="Green" title="E52">A138</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E19:29_1_1_4_4_2_1"/>
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1_4_4"><i><font color="Green" title="E48">A134</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1_4_4_2_1"><i><font color="Green" title="E52">A138</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c20">o</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="E2:29_1_1_4_4_2"/><i><font color="Green" title="E49">A148</font></i>: 
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</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="c19">k</font></span>)</span>
 ;<br/>

<a NAME="E3:29_1_1_4_4_2"/>
<font color="Maroon" title="c19">k</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="c19">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <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="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_4"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4"><i><font color="Green" title="E48">A134</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">u</font> = <font color="Maroon" title="c20">o</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1_4_4_2"><i><font color="Green" title="E49">A148</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">p0</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E6:29_1_1_4_4_2"/><i><font color="Green" title="E50">A149</font></i>: 
( <font color="Maroon" title="c22">p0</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="c4">b1</font> &amp; <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="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_3.html#T18" target="_self" title="PENCIL_3:th.18">Th18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font> as   <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> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_4_2"><i><font color="Green" title="E50">A149</font></i></a>, <a class="ref" href="pencil_1.html#T25" title="PENCIL_1:th.25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">fp</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14" title="PENCIL_1:th.14">PENCIL_1:14</a></i>;<br/>
<a NAME="E9:29_1_1_4_4_2"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>
<a NAME="E10:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon" title="c24">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">u</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_4_2"><i><font color="Green" title="E50">A149</font></i></a></i>;<br/>
<a NAME="E11:29_1_1_4_4_2"/><b>then </b><i><font color="Green" title="E51">A150</font></i>: 
<font color="Maroon" title="c13">M</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a></i>;<br/>
<a NAME="E12:29_1_1_4_4_2"/><i><font color="Green" title="E52">A151</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c24">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">b2</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="E13:29_1_1_4_4_2"/><i><font color="Green" title="E53">A152</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c22">p0</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_4_4_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_2"/><i><font color="Green" title="E54">A153</font></i>: 
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_4_4_2_2_1"/>
( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c25">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_2_1_1"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_4_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1_4_4"><i><font color="Green" title="E48">A134</font></i></a>, <a class="txt" href="pencil_3.html#E2:29_1_1_4_4_2"><i><font color="Green" title="E49">A148</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4_2"><i><font color="Green" title="E53">A152</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_4_4_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_4_2_2_1_2"/><i><font color="Green" title="E55">A154</font></i>: 
<font color="Maroon" title="c25">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_4_4_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_4_2"><i><font color="Green" title="E50">A149</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4_2"><i><font color="Green" title="E53">A152</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_4_2_2"><i><font color="Green" title="E54">A153</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_4_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p0</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<font color="Maroon" title="c21">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1_4_4"><i><font color="Green" title="E47">A133</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_4_4_2_2_1_2"><i><font color="Green" title="E55">A154</font></i></a>, <a class="ref" href="pencil_1.html#T26" title="PENCIL_1:th.26">PENCIL_1:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon" title="c23">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">K</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E6:29_1_1_4_4"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4_2"><i><font color="Green" title="E53">A152</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E16:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon" title="c24">fp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E17:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon" title="c24">fp</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="c9">b2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b2</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c9">b2</font></span>)</span>,<font color="Maroon" title="c16">S</font></span>)</span> <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="c9">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1_4_4_2"><i><font color="Green" title="E52">A151</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E18:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon" title="c24">fp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1_4_4_2"><i><font color="Green" title="E52">A151</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E19:29_1_1_4_4_2"/>
<font color="Maroon" title="c20">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">M</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a>, <a class="txt" href="pencil_3.html#E11:29_1_1_4_4_2"><i><font color="Green" title="E51">A150</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:29_1_1_4_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">M</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c16">S</font> is   <a href="pencil_1.html#NM1" title="PENCIL_1:NM.1">Block</a> of <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="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_4"><i><font color="Green" title="E41">A73</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1_4"><i><font color="Green" title="E44">A104</font></i></a>, <a class="txt" href="pencil_3.html#E6:29_1_1_4_4"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1_4_4"><i><font color="Green" title="E48">A134</font></i></a>, <a class="ref" href="funct_1.html#T155" title="FUNCT_1:th.155">FUNCT_1:155</a></i>;<br/>
<b>hence </b><a NAME="E16:29_1_1_4_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">M</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">S0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


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


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

<b>let </b><font color="Maroon" title="c15">a</font> be    <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="E24:29_1_1"/><i><font color="Green" title="E41">A155</font></i>: 
( <font color="Maroon" title="c15">a</font> is   <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> &amp; <font color="Maroon" title="c15">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="c4">b1</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c16">b</font> be    <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="E25:29_1_1"/><i><font color="Green" title="E42">A156</font></i>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font>
 ;<br/>

<a NAME="E26:29_1_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">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="E27:29_1_1"/><b>then </b>
<font color="Maroon" title="c15">a</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="c4">b1</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E16:29_1_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="txt" href="pencil_3.html#E24:29_1_1"><i><font color="Green" title="E41">A155</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E28:29_1_1"/><b>then </b>
<font color="Maroon" title="c15">a</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="c4">b1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">ai</font> = <font color="Maroon" title="c15">a</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="c4">b1</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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="c4">b1</font></span>)</span></span>)</span> ;<br/>
<a NAME="E30:29_1_1"/><i><font color="Green" title="E43">A157</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">b1</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_5"/><i><font color="Green" title="E44">A158</font></i>: 
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_5_1"/>
( <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> or <font color="Maroon" title="c18">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_5_1_1"/><i><font color="Green" title="E45">A159</font></i>: 
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E30:29_1_1"><i><font color="Green" title="E43">A157</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_5_1_1"/>
<font color="Maroon" title="c15">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1_5_1_1"><i><font color="Green" title="E45">A159</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_5_1_2"/>
<font color="Maroon" title="c18">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_5_1_2"/>
<font color="Maroon" title="c15">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E24:29_1_1"><i><font color="Green" title="E41">A155</font></i></a>, <a class="txt" href="pencil_3.html#E30:29_1_1"><i><font color="Green" title="E43">A157</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1_5"><i><font color="Green" title="E44">A158</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E32:29_1_1"/><b>then </b>
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1: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="c4">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E30:29_1_1"><i><font color="Green" title="E43">A157</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E33:29_1_1"/><b>then </b>
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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"><font color="Maroon" title="c4">b1</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">ai</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="pencil_3.html#E24:29_1_1"><i><font color="Green" title="E41">A155</font></i></a>, <a class="txt" href="pencil_3.html#E25:29_1_1"><i><font color="Green" title="E42">A156</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E34:29_1_1"/>
<font color="Maroon" title="c16">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" title="PENCIL_3:func.2">permutation_of_indices</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E20:29_1_1"><i><font color="Green" title="E40">A68</font></i></a></i>;<br/>


</div>
