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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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>, <font color="Olive" title="b2">g</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Olive" title="b2">g</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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">  <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">G</font> =  <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">FG</font> = <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">ONE</font> =  <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>;<br/>
<a NAME="E1:91"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span> <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="c2">f</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> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E2:91"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>;<br/>
<a NAME="E3:91"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>;<br/>
<a NAME="E4:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span> <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">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E5:91"/><span class="lab"><font color="Green" title="E64">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>;<br/>
<a NAME="E6:91"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <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="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<a NAME="E7:91"/><span class="lab"><font color="Green" title="E66">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span></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">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E67">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91_1"/><span class="lab"><font color="Green" title="E68">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E2:91_1"/><span class="lab"><font color="Green" title="E69">A10</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 ; <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="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span><br/><a NAME="E3:91_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E7:91"><span class="lab"><font color="Green" title="E66">A7</font></span></a>, <a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E2:91_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:91_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A11</font></span>: 
<span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>;<br/><a NAME="E5:91_1"/><span class="lab"><font color="Green" title="E71">A12</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E68">A9</font></span></a>, <a class="txt" href="#E2:91_1"><span class="lab"><font color="Green" title="E69">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_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="E1:91_1_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span> &amp; <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91_1"><span class="lab"><font color="Green" title="E71">A12</font></span></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A13</font></span>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 ; <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="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> = 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E64">A5</font></span></a>, <a class="txt" href="#E4:91_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1_1_1"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/> <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="91_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span> &amp; <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</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="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:91_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A14</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:91_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A15</font></span>: 
<font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">n</font>
 ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> = 
 <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:91_1"><span class="lab"><font color="Green" title="E70">A11</font></span></a>, <a class="txt" href="#E3:91_1_1_1_2"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E4:91_1_1_1_2"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c9">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_1_1_2"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="matrixj1.html#D8" target="_self" title="MATRIXJ1:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91_1_1_1_2"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="txt" href="#E4:91_1_1_1_2"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:91_1"/>
<span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p2">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="matrixj1.html#K14" title="MATRIXJ1:func.14">.</a> <font color="Maroon" title="c8">i</font>
 ; <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="E9:91"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:91"/>
 <a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="matrixj1.html#K15" title="MATRIXJ1:func.15">^</a> <span class="p1">(<span class="default"><a href="matrixj1.html#K23" title="MATRIXJ1:func.23">1.</a> <font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c3">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91"><span class="lab"><font color="Green" title="E65">A6</font></span></a>, <a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="txt" href="#E8:91"><span class="lab"><font color="Green" title="E67">A8</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</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>
