<?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/>
<b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <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>, <font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green">E349</font></i>: 
( <font color="Maroon">f1</font> is <a href="pencil_2.html#V1">isomorphic</a> &amp; ( for <font color="Olive">a</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">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="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> holds <br/>for <font color="Olive">b</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> holds <br/><font color="Olive">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">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">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>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green">E350</font></i>: 
( <font color="Maroon">f2</font> is <a href="pencil_2.html#V1">isomorphic</a> &amp; ( for <font color="Olive">a</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">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="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> holds <br/>for <font color="Olive">b</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> holds <br/><font color="Olive">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">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">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> ) )
 ;<br/>

<i><font color="Green">E351</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
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 href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x0</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"><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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_1_2_2"/><i><font color="Green">E353</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:29_1_2_2"/>
 <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</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/>;<br/><b>reconsider </b><font color="Maroon">P</font> = <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/><b>set </b><font color="Maroon">a</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">x0</font>;<br/><a NAME="E8:29_1_2_2"/><i><font color="Green">E354</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <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">x0</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">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">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_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2_2_1"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">e</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_2_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_2_2_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> or <font color="Maroon">e</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_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_2_2_1_1_1"/><i><font color="Green">E357</font></i>: 
<font color="Maroon">e</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_2_2_1_1_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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:29_1_2_2_1_1_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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</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_2_2_1_1_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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</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_2_2_1_1_1"/>
<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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <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_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_2_2_1_1_2"/>
<font color="Maroon">e</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_2_2_1_1_2"/><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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</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_2_2_1_1_2"/>
<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">x0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</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="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="E10:29_1_2_2"/><b>then </b><i><font color="Green">E358</font></i>: 
<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">x0</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/><b>reconsider </b><font color="Maroon">a1</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">x0</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">b</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</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_2_2"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<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">x0</font></span>)</span> <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">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>
 <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/><a NAME="E14:29_1_2_2"/><i><font color="Green">E360</font></i>: 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<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">x0</font></span>)</span> <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">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>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:29"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E15:29_1_2_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E16:29_1_2_2"/><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">x0</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">x</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E17:29_1_2_2"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29_1_2"/>
<font color="Maroon">f1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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