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

<b>set </b><font color="Maroon">s</font> =  <a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">i</font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>;<br/>
<a NAME="E1:29_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds <br/> ( <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> iff for <font color="Olive">B1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font><br/> for <font color="Olive">b1</font>, <font color="Olive">b2</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b1</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">b2</font> &amp;  <a href="pencil_1.html#K3">indx</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Olive">b2</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:29_1_1"/><i><font color="Green">E47</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:29_1_1"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E4:29_1_1"/><b>then </b>
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E5:29_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E6:29_1_1"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a></i>;<br/>
<a NAME="E7:29_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E8:29_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="pencil_2.html#T24">PENCIL_2:24</a></i>;<br/>
<b>consider </b><font color="Maroon">b2</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E12:29_1_1"/><i><font color="Green">E53</font></i>: 
(  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/>
<a NAME="E13:29_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, <a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">j</font> =  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>;<br/>
<a NAME="E14:29_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T146">FUNCT_1:146</a></i>;<br/>
<a NAME="E15:29_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T152">FUNCT_1:152</a></i>;<br/>
<a NAME="E16:29_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">J</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">set</a>  = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>;<br/>
<a NAME="E17:29_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pencil_1.html#K4">Segre_Blocks</a> <font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pencil_1.html#D23">PENCIL_1:def 23</a></i>;<br/>
<a NAME="E18:29_1_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<b>consider </b><font color="Maroon">bb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">bf</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:29_1_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; ( for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> holds <br/><font color="Maroon">bf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:29_1_1_1"/><i><font color="Green">E104</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">bbx</font> = <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<i><font color="Green">E105</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><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">u</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E107</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, </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">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_1_2_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E4:29_1_1_1_2_1_2"/>
<font color="Maroon">b1</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E5:29_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:29_1_1_1"/><b>then </b>
<font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">bbx1</font> = <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E12:29_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">bbx1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:29_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</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">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1_3_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_1_3_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E111</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/><a NAME="E2:29_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:29_1_1_1_3_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">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">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E140</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_7.html#T33">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">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">E141</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_3.html#T18">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">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">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:29_1_1_1_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:29_1_1_1_3"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green">E144</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> = 
<font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_3_3"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E2:29_1_1_1_3_3"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E4:29_1_1_1_3_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</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">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E148</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_1_3_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_3_3_1_1"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T33">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">E149</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_3_3_1_2"/><b>then </b>
( not <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_1_3_3_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E5:29_1_1_1_3_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E6:29_1_1_1_3_3_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T34">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, </i>;<br/><b>hence </b><a NAME="E9:29_1_1_1_3_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D1">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">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1_1_1_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fbbx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">bbx1</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E16:29_1_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">fbbx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">y1</font> = 
<font color="Maroon">fbbx</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i>, <a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E19:29_1_1"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>() holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:29_1_1_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>()
 ;<br/>

<b>consider </b><font color="Maroon">bb</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">bb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">bf</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:29_1_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bf</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> &amp; ( for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> holds <br/><font color="Maroon">bf</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:29_1_1_2"/><i><font color="Green">E160</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">bb</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">bbx</font> = <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>;<br/>
<i><font color="Green">E161</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><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">u</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E163</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></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">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E4:29_1_1_2_2_1_2"/>
<font color="Maroon">b1</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/><a NAME="E5:29_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:29_1_1_2"/><b>then </b>
<font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">bbx1</font> = <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E12:29_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:29_1_1_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">bbx1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E14:29_1_1_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</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">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_2_3_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:29_1_1_2_3_1"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E167</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/><a NAME="E2:29_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:29_1_1_2_3_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">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">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E170</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T33">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">E171</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="card_3.html#T18">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T34">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">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:29_1_1_2_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:29_1_1_2_3"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green">E173</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_2_3_3"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E2:29_1_1_2_3_3"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><a NAME="E4:29_1_1_2_3_3"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</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">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E179</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_2_3_3_1_1"/><b>then </b>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_2_3_3_1_1"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="funct_7.html#T33">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">E180</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2_3_3_1_2"/><b>then </b>
( not <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">zz</font> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/><b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_2_3_3_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><a NAME="E5:29_1_1_2_3_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E6:29_1_1_2_3_3_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D1">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">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">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E9:29_1_1_2_3_3_1_2"/>
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tarski.html#D1">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">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:29_1_1_2_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:29_1_1_2"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K2">+*</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fbbx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">bbx1</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> = <font color="Maroon">fbbx</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>;<br/>

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

<b>assume </b><a NAME="E17:29_1_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E18:29_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">M</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:29_1_1"/><i><font color="Green">E185</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>() &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>set </b><font color="Maroon">t</font> =  <a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font>;<br/>
<a NAME="E22:29_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">o</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_3"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E6:29_1_1_3"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E7:29_1_1_3"/>
<font color="Maroon">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">pio</font> = <font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<a NAME="E9:29_1_1_3"/>
<font color="Maroon">pio</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:29_1_1_3"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pio</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">pio</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E12:29_1_1_3"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:29_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E14:29_1_1_3"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:29_1_1_3"/>
<span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
;<br/>
<a NAME="E16:29_1_1_3"/><b>then </b><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:29_1_1_3"/>
<font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E18:29_1_1_3"/><b>then </b>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E19:29_1_1_3"/><b>then </b>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E20:29_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:29_1_1"/><b>then </b>
<font color="Maroon">M</font> is    <a href="relset_1.html#M2">Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>set </b><font color="Maroon">m</font> = <font color="Maroon">M</font>;<br/>
<b>thus </b><a NAME="E25:29_1_1"/>
<font color="Maroon">M</font> is <a href="pencil_2.html#V1">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">E192</font></i>: 
<font color="Maroon">M</font> is <a href="funct_1.html#V2">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">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">o1</font> = <font color="Maroon">x1</font>, <font color="Maroon">o2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:29_1_1_4_1"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E5:29_1_1_4_1"/>
<font color="Maroon">p1</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p1io</font> = <font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<a NAME="E7:29_1_1_4_1"/>
<font color="Maroon">p1io</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:29_1_1_4_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1io</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1io</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E10:29_1_1_4_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">J1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:29_1_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o1</font>,<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E12:29_1_1_4_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">p2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E14:29_1_1_4_1"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E15:29_1_1_4_1"/>
<font color="Maroon">p2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p2io</font> = <font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<a NAME="E17:29_1_1_4_1"/>
<font color="Maroon">p2io</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:29_1_1_4_1"/><b>then </b><i><font color="Green">E245</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2io</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<b>reconsider </b><font color="Maroon">J2</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2io</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E20:29_1_1_4_1"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">J2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">o2</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E21:29_1_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">o2</font>,<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E22:29_1_1_4_1"/><b>then </b><i><font color="Green">E248</font></i>: 
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:29_1_1_4_1"/><i><font color="Green">E249</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E251</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">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">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:29_1_1_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</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_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_4_1_1_1_2"/>
<font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</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_1"/><b>then </b>
<font color="Maroon">p1io</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E26:29_1_1_4_1"/><b>then </b><i><font color="Green">E252</font></i>: 
<font color="Maroon">J1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E27:29_1_1_4_1"/><i><font color="Green">E253</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E255</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_7.html#T33">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">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:29_1_1_4_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_4_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E29:29_1_1_4_1"/><b>then </b>
<font color="Maroon">p2io</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E30:29_1_1_4_1"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">J2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E31:29_1_1_4_1"/><i><font color="Green">E257</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_1_3"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">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">l</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> or <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">l</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_1_3_1_1"/>
<font color="Maroon">J1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</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">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_4_1_3_1_2"/>
<font color="Maroon">J1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pencil_1.html#T23">PENCIL_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E33:29_1_1_4_1"/><b>then </b>
<font color="Maroon">J1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">J2</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E34:29_1_1_4_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<font color="Maroon">p1io</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2io</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E35:29_1_1_4_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E36:29_1_1_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E37:29_1_1_4_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29_1_1_4"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">M</font> is <a href="funct_2.html#V2">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">rng</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3">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">rng</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">p0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:29_1_1_4_2_1_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">x1</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E5:29_1_1_4_2_1_1"/>
<font color="Maroon">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E8:29_1_1_4_2_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E10:29_1_1_4_2_1_1"/><i><font color="Green">E262</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:29_1_1_4_2_1_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:29_1_1_4_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E13:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E14:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:29_1_1_4_2_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E16:29_1_1_4_2_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:29_1_1_4_2_1_1"/><i><font color="Green">E266</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:29_1_1_4_2_1_1"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:29_1_1_4_2_1_1"/><i><font color="Green">E268</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_2_1_1_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">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">l</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">l</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">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">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E270</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><a NAME="E3:29_1_1_4_2_1_1_1_1_2"/><i><font color="Green">E271</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_2_1_1_1_1_2_1"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E273</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T33">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">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E4:29_1_1_4_2_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_4_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#T18">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">p</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E6:29_1_1_4_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">q0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:29_1_1_4_2_1_1_1_1_2"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">q0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:29_1_1_4_2_1_1_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q0</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E10:29_1_1_4_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><b>hence </b><a NAME="E11:29_1_1_4_2_1_1_1_1_2"/>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:29_1_1_4_2_1_1"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E22:29_1_1_4_2_1_1"/>
<font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E23:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">x1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E24:29_1_1_4_2_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E25:29_1_1_4_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T12">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">E276</font></i>: 
<font color="Maroon">M</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pencil_2.html#D4">PENCIL_2:def 4</a><br/>

<a NAME="E4:29_1_1_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:29_1_1_4"/><b>then </b><i><font color="Green">E277</font></i>: 
<font color="Maroon">M</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>thus </b><a NAME="E6:29_1_1_4"/>
<font color="Maroon">M</font> is <a href="t_0topsp.html#V1">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">S0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a><br/>

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

<a NAME="E2:29_1_1_4_3"/><b>then </b>
<font color="Maroon">S0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S0</font> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">f</font> <a href="pencil_2.html#K2">.:</a> <font color="Maroon">L</font> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">k</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:29_1_1_4_3"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24">PENCIL_1:24</a></i>;<br/>
<a NAME="E8:29_1_1_4_3"/><i><font color="Green">E281</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:29_1_1_4_3"/><i><font color="Green">E282</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E283</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_7.html#T33">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E4:29_1_1_4_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_4_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></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">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>
<a NAME="E12:29_1_1_4_3"/><b>then </b>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E13:29_1_1_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">b2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:29_1_1_4_3"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">b2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E15:29_1_1_4_3"/><b>then </b><i><font color="Green">E285</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> holds <br/><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#T26">PENCIL_1:26</a></i>;<br/>
<a NAME="E16:29_1_1_4_3"/>
<font color="Maroon">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_3_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_4_3_2_1"/><i><font color="Green">E286</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">p0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E6:29_1_1_4_3_2_1"/><i><font color="Green">E287</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, , <a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E9:29_1_1_4_3_2_1"/><i><font color="Green">E288</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_1_1"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E290</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_1_4_3_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_4_3_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T33">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">E291</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#T18">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T34">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">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E12:29_1_1_4_3_2_1"/><b>then </b><i><font color="Green">E292</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E13:29_1_1_4_3_2_1"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E14:29_1_1_4_3_2_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">u</font>,<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E15:29_1_1_4_3_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E16:29_1_1_4_3_2_1"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E17:29_1_1_4_3_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:29_1_1_4_3_2"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E3:29_1_1_4_3_2"/>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">p0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E7:29_1_1_4_3_2"/><i><font color="Green">E296</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">k</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:29_1_1_4_3_2"/>
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E9:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E10:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E13:29_1_1_4_3_2"/><i><font color="Green">E297</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E14:29_1_1_4_3_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E15:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E16:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:29_1_1_4_3_2"/><b>then </b><i><font color="Green">E298</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:29_1_1_4_3_2"/><i><font color="Green">E299</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_2"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_3_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_3_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></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">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E20:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="pencil_2.html#K2">.:</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">qq</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:29_1_1_4_3_2"/><i><font color="Green">E303</font></i>: 
( <font color="Maroon">qq</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">qq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">qq</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E23:29_1_1_4_3_2"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>;<br/>
<a NAME="E24:29_1_1_4_3_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:29_1_1_4_3_2"/><b>then </b><i><font color="Green">E305</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E26:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E27:29_1_1_4_3_2"/><b>then </b><i><font color="Green">E306</font></i>: 
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">qi</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">q0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E30:29_1_1_4_3_2"/><i><font color="Green">E308</font></i>: 
( <font color="Maroon">q0</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">qi</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E31:29_1_1_4_3_2"/><i><font color="Green">E309</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_3_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_7.html#T33">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">E310</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="pencil_3.html#T6" target="_self">Th6</a>, , , <a class="ref" href="pencil_1.html#T23">PENCIL_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E33:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">q0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">qi</font> <a href="pboole.html#R6">=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E34:29_1_1_4_3_2"/><b>then </b><i><font color="Green">E311</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">qi</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E35:29_1_1_4_3_2"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">qi</font>,<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">qi</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E36:29_1_1_4_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">qi</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E37:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E38:29_1_1_4_3_2"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<b>hence </b><a NAME="E39:29_1_1_4_3_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pencil_3.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:29_1_1_4_3"/><b>then </b>
<font color="Maroon">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E18:29_1_1_4_3"/>
<font color="Maroon">M</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">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">M</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i>, <a class="ref" href="pencil_2.html#T12">PENCIL_2:12</a></i>;<br/>

<b>thus </b><a NAME="E8:29_1_1_4"/>
<font color="Maroon">M</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="t_0topsp.html#V1">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">S0</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D2">T_0TOPSP:def 2</a><br/>

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

<a NAME="E2:29_1_1_4_4"/><b>then </b>
<font color="Maroon">S0</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S0</font> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">f</font> <a href="pencil_2.html#K3">"</a> <font color="Maroon">L</font> as   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">k</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E7:29_1_1_4_4"/><i><font color="Green">E313</font></i>: 
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_1.html#T24">PENCIL_1:24</a></i>;<br/>
<a NAME="E8:29_1_1_4_4"/><i><font color="Green">E314</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E315</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_7.html#T33">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">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E4:29_1_1_4_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_4_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></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">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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_4"/><b>then </b>
<font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_3.html#T38">CARD_3:38</a></i>;<br/>
<a NAME="E11:29_1_1_4_4"/><b>then </b>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E1:29_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<a NAME="E12:29_1_1_4_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E13:29_1_1_4_4"/><b>then </b>
2 <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">b1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D6">PENCIL_1:def 6</a></i>;<br/>
<a NAME="E14:29_1_1_4_4"/><b>then </b><i><font color="Green">E316</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> &amp; ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> holds <br/><font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> ) )
 
<b>by </b><i><a class="ref" href="pencil_1.html#T26">PENCIL_1:26</a></i>;<br/>
<a NAME="E15:29_1_1_4_4"/>
<font color="Maroon">M</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">M</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">o</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_4_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green">E317</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:29_1_1_4_4_2_1"/><i><font color="Green">E318</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, , <a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fq</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E8:29_1_1_4_4_2_1"/><i><font color="Green">E320</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:29_1_1_4_4_2_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E10:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green">E321</font></i>: 
<font color="Maroon">fq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:29_1_1_4_4_2_1"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green">E322</font></i>: 
<font color="Maroon">fq</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E13:29_1_1_4_4_2_1"/>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">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">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_4_4_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:29_1_1_4_4_2_1_1"/><i><font color="Green">E323</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp; ( for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:29_1_1_4_4_2_1_1"/><i><font color="Green">E324</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pboole.html#D3">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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_1_1_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:29_1_1_4_4_2_1_1_1"/><b>then </b><i><font color="Green">E325</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E326</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_7.html#T33">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">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:29_1_1_4_4_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E5:29_1_1_4_4_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/><b>hence </b><a NAME="E6:29_1_1_4_4_2_1_1_1_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i>, , </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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_4_2_1_1_1_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T4" target="_self">Th4</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">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29_1_1_4_4_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E15:29_1_1_4_4_2_1"/><b>then </b><i><font color="Green">E327</font></i>: 
<font color="Maroon">fq</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E16:29_1_1_4_4_2_1"/><i><font color="Green">E328</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_1_2"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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"/><i><font color="Green">E330</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_4_2_1_2_1_1"/>
<font color="Maroon">fq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">fq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="card_3.html#T18">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_4_2_1"/><b>then </b>
<font color="Maroon">fq</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E19:29_1_1_4_4_2_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E20:29_1_1_4_4_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E21:29_1_1_4_4_2_1"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:29_1_1_4_4_2"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>

<a NAME="E3:29_1_1_4_4_2"/>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E4:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">o</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">p0</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E7:29_1_1_4_4_2"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">p0</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:29_1_1_4_4_2"/>
<font color="Maroon">p0</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="pencil_1.html#T25">PENCIL_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fp</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pencil_1.html#T14">PENCIL_1:14</a></i>;<br/>
<a NAME="E11:29_1_1_4_4_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T117">FUNCT_1:117</a></i>;<br/>
<a NAME="E12:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">fp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">u</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E13:29_1_1_4_4_2"/><b>then </b><i><font color="Green">E334</font></i>: 
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E14:29_1_1_4_4_2"/><i><font color="Green">E335</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:29_1_1_4_4_2"/><i><font color="Green">E336</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_4_4_2_2"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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"/><i><font color="Green">E338</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_4_4_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_4_4_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E7:29_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></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">E339</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="pencil_3.html#E13:29_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="card_3.html#T18">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">p0</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E14:29_1_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E17:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E12:29_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E18:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">fp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E19:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">fp</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E20:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">fp</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:29"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E21:29_1_1_4_4_2"/><b>then </b>
<font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:29_1_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:29_1_1_4_4_2"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:29_1_1_4_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S</font> is   <a href="pencil_1.html#NM1">Block</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , , <a class="txt" href="pencil_3.html#E5:29_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T155">FUNCT_1:155</a></i>;<br/>
<b>hence </b><a NAME="E17:29_1_1_4_4"/>
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">S0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


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


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

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

<b>assume </b><a NAME="E26:29_1_1"/><i><font color="Green">E340</font></i>: 
( <font color="Maroon">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> )
 ;<br/>

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

<b>assume </b><a NAME="E27:29_1_1"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E28:29_1_1"/><i><font color="Green">E342</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E29:29_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E30:29_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E31:29_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">ai</font> = <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> ;<br/>
<a NAME="E33:29_1_1"/><i><font color="Green">E344</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<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">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_5"/><i><font color="Green">E345</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">E346</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_5_1_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:29_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D1">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">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">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">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_5_1_2"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E35:29_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E36:29_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K2">+*</a> <span class="p3">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span>,<span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">ai</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E37:29_1_1"/>
<font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K2" target="_self">permutation_of_indices</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
