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<b>let </b><font color="Maroon">s</font> be   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">I</font>, <font color="Maroon">t</font> be   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">I</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:28_1_1"/><i><font color="Green">E35</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/> ( <font color="Maroon">s</font> <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>and </b><br/><a NAME="E2:28_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/> ( <font color="Maroon">t</font> <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> )
 ;<br/>

<a NAME="E3:28_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</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="28_1_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="E1:28_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>reconsider </b><font color="Maroon">j2</font> = <font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">b1</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:28_1_1_1"/><i><font color="Green">E48</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">B1</font> =  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">fB</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="E10:28_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">fB</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</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/><font color="Maroon">j1</font> = 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">j2</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E12:28_1_1_1"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:28_1_1"/>
<font color="Maroon">s</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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