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

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">H</font> being   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b3">FG</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b4">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">H</font><br/> for <font color="Olive" title="b5">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b3">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">FG</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="Olive" title="b5">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b5">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">FG</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b3">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H</font><br/> for <font color="Olive" title="b4">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b2">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">FG</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="Olive" title="b4">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b4">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">H</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FG</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><br/> for <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">FG</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="Olive" title="b3">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">FG</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><br/> for <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">FG</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="Olive" title="b2">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">FH</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">FG</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="Olive" title="b1">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">FG</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="c6">I</font> implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> )</span><br/>











<b>assume </b><a NAME="E1:143"/><i><font color="Green" title="E119">A1</font></i>: 
( <font color="Maroon" title="c4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">FG</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="c6">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">FG</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><br/> for <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">FG</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="Olive" title="b3">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span>;<br/>
<a NAME="E2:143"/><i><font color="Green" title="E120">A2</font></i>: 
<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="143_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">FG</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><br/> for <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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="Olive" title="b2">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">FH</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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="Olive" title="b1">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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="c9">I</font> implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:143_1"/><i><font color="Green" title="E121">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FH</font> or  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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="c9">I</font> or  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> )</span><br/>

<a NAME="E2:143_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>;<br/>
<a NAME="E3:143_1"/><b>then </b>
<font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 
;<br/>
<a NAME="E4:143_1"/><b>then </b><i><font color="Green" title="E122">A4</font></i>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a></i>;<br/>
<b>assume </b><a NAME="E5:143_1"/>
( <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">FG</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="c9">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span></span><br/>

<a NAME="E6:143_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:143_1"><i><font color="Green" title="E121">A3</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>;<br/>
<a NAME="E7:143_1"/><b>then </b>
<font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>
 
;<br/>
<a NAME="E8:143_1"/><b>then </b>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a></i>;<br/>
<b>hence </b><a NAME="E9:143_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:143_1"><i><font color="Green" title="E122">A4</font></i></a>, <a class="ref" href="group_9.html#T4" target="_self" title="GROUP_9:th.4">Th4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E121">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="143_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:143_2"/><i><font color="Green" title="E122">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><b>thus </b><a NAME="E2:143_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">FG</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FH</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font><br/> for <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="Olive" title="b2">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">FH</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="Olive" title="b1">I</font> holds <br/> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b1">I</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c10">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="c10">I</font> implies  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> )</span><br/>





<b>assume </b><a NAME="E1:143_2_1"/><i><font color="Green" title="E123">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FH</font> or  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="c10">I</font> or  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:143_2_1"/><i><font color="Green" title="E124">A8</font></i>: 
( <font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FH</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="c10">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c11">In</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E4:143_2_1"/><i><font color="Green" title="E125">A9</font></i>: 
<font color="Maroon" title="c10">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">In</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">FGn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c14">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E6:143_2_1"/><i><font color="Green" title="E126">A10</font></i>: 
<font color="Maroon" title="c8">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FGn</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">FHn</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font>, <font color="Maroon" title="c16">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font><b> such that </b><br/><a NAME="E8:143_2_1"/><i><font color="Green" title="E127">A11</font></i>: 
<font color="Maroon" title="c9">FH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FHn</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">I1</font> = <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c18">FG1</font> = <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon" title="c19">FH1</font> = <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>;<br/>
<a NAME="E9:143_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">I</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="c11">In</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="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:143_2_1"><i><font color="Green" title="E125">A9</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E10:143_2_1"/><b>then </b><i><font color="Green" title="E128">A12</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c11">In</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E11:143_2_1"/><i><font color="Green" title="E129">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">FG</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="c13">FGn</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="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:143_2_1"><i><font color="Green" title="E126">A10</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E12:143_2_1"/><b>then </b><i><font color="Green" title="E130">A14</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">FGn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E13:143_2_1"/><i><font color="Green" title="E131">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">FH</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="c15">FHn</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="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:143_2_1"><i><font color="Green" title="E127">A11</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E14:143_2_1"/><b>then </b><i><font color="Green" title="E132">A16</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c15">FHn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<i><font color="Green" title="E133">A17</font></i>: <font color="Maroon" title="c8">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">FGn</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">FGn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="#E6:143_2_1"><i><font color="Green" title="E126">A10</font></i></a>, <a class="txt" href="#E11:143_2_1"><i><font color="Green" title="E129">A13</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c14">g</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/>
<i><font color="Green" title="E134">A18</font></i>: <font color="Maroon" title="c9">FH</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">FHn</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">FHn</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:143_2_1"><i><font color="Green" title="E123">A7</font></i></a>, <a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="txt" href="#E8:143_2_1"><i><font color="Green" title="E127">A11</font></i></a>, <a class="txt" href="#E13:143_2_1"><i><font color="Green" title="E131">A15</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c16">h</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/>
<a NAME="E17:143_2_1"/><b>then </b>
<font color="Maroon" title="c13">FGn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FHn</font>
 
<b>by </b><i><a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="txt" href="#E6:143_2_1"><i><font color="Green" title="E126">A10</font></i></a>, <a class="txt" href="#E8:143_2_1"><i><font color="Green" title="E127">A11</font></i></a>, <a class="txt" href="#E15:143_2_1"><i><font color="Green" title="E133">A17</font></i></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>
<a NAME="E18:143_2_1"/><b>then </b><i><font color="Green" title="E135">A19</font></i>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">FGn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">FHn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:143_2"><i><font color="Green" title="E122">A6</font></i></a>, <a class="txt" href="#E10:143_2_1"><i><font color="Green" title="E128">A12</font></i></a>, <a class="txt" href="#E14:143_2_1"><i><font color="Green" title="E132">A16</font></i></a></i>;<br/>
<i><font color="Green" title="E136">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> = 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<i><font color="Green" title="E137">A21</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> = 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
<i><font color="Green" title="E138">A22</font></i>:  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">FGn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:143_2_1"><i><font color="Green" title="E125">A9</font></i></a>, <a class="txt" href="#E6:143_2_1"><i><font color="Green" title="E126">A10</font></i></a>, <a class="txt" href="#E10:143_2_1"><i><font color="Green" title="E128">A12</font></i></a>, <a class="txt" href="#E12:143_2_1"><i><font color="Green" title="E130">A14</font></i></a>, <a class="txt" href="#E19:143_2_1"><i><font color="Green" title="E136">A20</font></i></a>, <a class="ref" href="group_4.html#T25" title="GROUP_4:th.25">GROUP_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">FGn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
;<br/>
<div><i><font color="Green" title="E139">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="143_2_1_6"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c20">H'</font> = <font color="Maroon" title="c3">H</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">h'</font> = <font color="Maroon" title="c16">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c20">H'</font> ;<br/><a NAME="E3:143_2_1_6"/>
<font color="Maroon" title="c14">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">h'</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="#E2:143_2_1"><i><font color="Green" title="E124">A8</font></i></a>, <a class="txt" href="#E15:143_2_1"><i><font color="Green" title="E133">A17</font></i></a>, <a class="txt" href="#E16:143_2_1"><i><font color="Green" title="E134">A18</font></i></a>, <a class="ref" href="group_4.html#T4" title="GROUP_4:th.4">GROUP_4:4</a></i>;<br/><b>hence </b><a NAME="E4:143_2_1_6"/>
<font color="Maroon" title="c14">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c14">g</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>

<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="group_4.html#T28" title="GROUP_4:th.28">GROUP_4:28</a></i>
<br/>.= 
<font color="Maroon" title="c16">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="txt" href="#E22:143_2_1"><i><font color="Green" title="E139">A23</font></i></a>, <a class="ref" href="group_4.html#T12" title="GROUP_4:th.12">GROUP_4:12</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="group_4.html#T12" title="GROUP_4:th.12">GROUP_4:12</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T28" title="GROUP_4:th.28">GROUP_4:28</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span>

;<br/>
<b>then </b> <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">FHn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E18:143_2_1"><i><font color="Green" title="E135">A19</font></i></a>, <a class="txt" href="#E21:143_2_1"><i><font color="Green" title="E138">A22</font></i></a>, <a class="ref" href="group_9.html#T3" target="_self" title="GROUP_9:th.3">Th3</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FHn</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c11">In</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">FHn</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c16">h</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">In</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c12">i</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:143_2_1"><i><font color="Green" title="E128">A12</font></i></a>, <a class="txt" href="#E14:143_2_1"><i><font color="Green" title="E132">A16</font></i></a>, <a class="txt" href="#E20:143_2_1"><i><font color="Green" title="E137">A21</font></i></a>, <a class="ref" href="group_4.html#T25" title="GROUP_4:th.25">GROUP_4:25</a></i>
;<br/>
<b>hence </b><a NAME="E25:143_2_1"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c10">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:143_2_1"><i><font color="Green" title="E125">A9</font></i></a>, <a class="txt" href="#E8:143_2_1"><i><font color="Green" title="E127">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:143"/><i><font color="Green" title="E122">A24</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:143"><i><font color="Green" title="E120">A2</font></i></a>, <a class="txt" href="#E3:143"><i><font color="Green" title="E121">A5</font></i></a>);<br/></i>
<b>thus </b><a NAME="E5:143"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">FG</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">FH</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:143"><i><font color="Green" title="E119">A1</font></i></a>, <a class="txt" href="#E4:143"><i><font color="Green" title="E122">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
