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

<span class="kw">let </span><font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b3">A</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b2">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Olive" title="b3">A</font> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>  st <font color="Olive" title="b1">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Olive" title="b2">A</font> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b2">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>  st <font color="Maroon" title="c4">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Olive" title="b1">A</font> holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">E</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Maroon" title="c5">A</font> implies <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:93"/><span class="lab"><font color="Green" title="E71">A1</font></span>: 
<font color="Maroon" title="c4">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:93_1"/>
( <font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_1"><span class="kw">suppose </span></a><a NAME="E1:93_1_1"/>
<font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:93_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:93_1_1"/><span class="kw">then </span>
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D3" target="_self" title="GROUP_9:def.3">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:93_1_1"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="93_1_2"><span class="kw">suppose </span></a><a NAME="E1:93_1_2"/><span class="lab"><font color="Green" title="E72">A2</font></span>: 
<font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>;<br/><a NAME="E2:93_1_2"/><span class="lab"><font color="Green" title="E73">A3</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_1_2_1"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:93_1_2_1_1"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">Fk</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>, <font color="Maroon" title="c9">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font><span class="kw"> such that </span><br/><a NAME="E3:93_1_2_1_1"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Fk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">o</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></span>;<br/><a NAME="E4:93_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">Fk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">o</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_1_2_1_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E5:93_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">Fk</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/><a NAME="E6:93_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A7</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">Fk</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E74">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">F1</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c9">o</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:93_1_2_1_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></span>;<br/><a NAME="E8:93_1_2_1_1"/>
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">Fk</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="#E3:93_1_2_1_1"><span class="lab"><font color="Green" title="E76">A6</font></span></a>, <a class="txt" href="group_9.html#E83"><span class="lab"><font color="Green" title="E69">Lm29</font></span></a></span>;<br/><a NAME="E9:93_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A8</font></span>: 
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">Fk</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></span>;<br/><a NAME="E10:93_1_2_1_1"/>
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="txt" href="group_9.html#E80"><span class="lab"><font color="Green" title="E66">Lm26</font></span></a></span>;<br/><a NAME="E11:93_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E71">A1</font></span></a>, <a class="txt" href="#E1:93_1_2"><span class="lab"><font color="Green" title="E72">A2</font></span></a>, <a class="ref" href="group_9.html#D1" target="_self" title="GROUP_9:def.1">Def1</a></span>;<br/><a NAME="E12:93_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c8">Fk</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93_1_2_1_1"><span class="lab"><font color="Green" title="E78">A8</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/><span class="kw">hence </span><a NAME="E13:93_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93_1_2_1_1"><span class="lab"><font color="Green" title="E77">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:93_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c6">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E4:93_1_2"/><span class="lab"><font color="Green" title="E74">A9</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F</font>
 ;<br/><a NAME="E5:93_1_2"/><span class="lab"><font color="Green" title="E75">A10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:93_1_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E2:93_1_2_2"/><span class="kw">then </span>
 <a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D3" target="_self" title="GROUP_9:def.3">Def3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:93_1_2_2"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:93_1_2"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E5:93_1_2"><span class="lab"><font color="Green" title="E75">A10</font></span></a>, <a class="txt" href="#E2:93_1_2"><span class="lab"><font color="Green" title="E73">A3</font></span></a>);<br/></span><span class="kw">hence </span><a NAME="E7:93_1_2"/>
<span class="p1">(<span class="default"><a href="group_9.html#K2" title="GROUP_9:func.2">Product</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c5">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93_1_2"><span class="lab"><font color="Green" title="E74">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
