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

<span class="kw">set </span><font color="Maroon" title="c1">f1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">f2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E2:15_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E3:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p3"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  )</span><br/>

<a NAME="E5:15_1_1"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
1 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A5</font></span>: <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T18" title="FINSEQ_4:th.18">FINSEQ_4:18</a></span>
;<br/>
<a NAME="E7:15_1_1"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
2 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E20">A7</font></span>: <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T18" title="FINSEQ_4:th.18">FINSEQ_4:18</a></span>
;<br/>
<a NAME="E9:15_1_1"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T26" title="FINSEQ_1:th.26">FINSEQ_1:26</a></span>;<br/>
<a NAME="E10:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E11:15_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E9:15_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E12:15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E10:15_1_1"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:15_1_1"/>
 not <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E7:15_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E9:15_1_1"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  )</span><br/>

<a NAME="E14:15_1_1"/>
3 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T18" title="FINSEQ_4:th.18">FINSEQ_4:18</a></span>
;<br/>
<a NAME="E16:15_1_1"/>
1 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
;<br/>
<a NAME="E18:15_1_1"/>
2 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E25">A12</font></span>: <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="finseq_4.html#T69" title="FINSEQ_4:th.69">FINSEQ_4:69</a></span>
<br/>.= 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
;<br/>
<a NAME="E20:15_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
;<br/>
<a NAME="E21:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A13</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E22:15_1_1"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
;<br/>
<a NAME="E23:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E24:15_1_1"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E25:15_1_1"/>
4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 
;<br/>
<a NAME="E26:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,4) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E27:15_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> or  not <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <font color="Maroon" title="c3">f</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <font color="Maroon" title="c3">f</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:15_1_1_6"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E3:15_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:15_1_1_6"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E5:15_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A18</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_6"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_6_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="E1:15_1_1_6_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_6"><span class="lab"><font color="Green" title="E31">A18</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_6_1_1"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_6_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1_1"><span class="lab"><font color="Green" title="E32">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_6_1_2"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_6_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_6_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_6_1_3"/><span class="lab"><font color="Green" title="E32">A21</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_6_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1_3"><span class="lab"><font color="Green" title="E32">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_6_1_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_6_1_4"/><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_6_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_6_1_4"><span class="lab"><font color="Green" title="E32">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">thus </span><a NAME="E28:15_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> or  not <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> <font color="Maroon" title="c3">f</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> <font color="Maroon" title="c3">f</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<a NAME="E2:15_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A23</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E3:15_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A24</font></span>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:15_1_1_7"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_7"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:15_1_1_7"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <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> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E6:15_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A26</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 or <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:15_1_1_7"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="txt" href="#E4:15_1_1_7"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_7_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="E1:15_1_1_7_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1_7"><span class="lab"><font color="Green" title="E33">A26</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_7_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E23:15_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_7_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E24:15_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="topreal1.html#T18" title="TOPREAL1:th.18">TOPREAL1:18</a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_7_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_7_1_3"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_7_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E26:15_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="topreal1.html#T16" title="TOPREAL1:th.16">TOPREAL1:16</a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">thus </span><a NAME="E29:15_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E30:15_1_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="goboard5.html#D4" target="_self" title="GOBOARD5:def.4">GOBOARD5:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; ( ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) or <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Olive" title="b1">j</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> &amp; ( ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15_1_1_8"/><span class="lab"><font color="Green" title="E30">A27</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E2:15_1_1_8"/><span class="lab"><font color="Green" title="E31">A28</font></span>: 
( ( <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> ) or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span><br/>

<a NAME="E3:15_1_1_8"/><span class="lab"><font color="Green" title="E32">A29</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E4:15_1_1_8"/><span class="lab"><font color="Green" title="E33">A30</font></span>: 
( <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_8"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/>
<a NAME="E5:15_1_1_8"/><span class="lab"><font color="Green" title="E34">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E6:15_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A32</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:15_1_1_8"/><span class="lab"><font color="Green" title="E36">A33</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
;<br/>
<a NAME="E8:15_1_1_8"/><span class="lab"><font color="Green" title="E37">A34</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
;<br/>
<a NAME="E9:15_1_1_8"/><span class="lab"><font color="Green" title="E38">A35</font></span>: 
( <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 implies <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
;<br/>
<a NAME="E10:15_1_1_8"/><span class="lab"><font color="Green" title="E39">A36</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E34">A31</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_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="E1:15_1_1_8_1_1"/>
( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E3:15_1_1_8"><span class="lab"><font color="Green" title="E32">A29</font></span></a>, <a class="txt" href="#E4:15_1_1_8"><span class="lab"><font color="Green" title="E33">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_1_1_1"><span class="kw">case </span></a><a NAME="E1:15_1_1_8_1_1_1"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_8_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="txt" href="#E9:15_1_1_8"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_8_1_1_2"><span class="kw">case </span></a><a NAME="E1:15_1_1_8_1_1_2"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_8_1_1_2"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_8"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="txt" href="#E5:15_1_1_8"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="txt" href="#E8:15_1_1_8"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="txt" href="#E9:15_1_1_8"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="ref" href="nat_1.html#T27" title="NAT_1:th.27">NAT_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_8_1_1_3"><span class="kw">case </span></a><a NAME="E1:15_1_1_8_1_1_3"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_8_1_1_3"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:15_1_1_8"><span class="lab"><font color="Green" title="E31">A28</font></span></a>, <a class="txt" href="#E6:15_1_1_8"><span class="lab"><font color="Green" title="E35">A32</font></span></a>, <a class="txt" href="#E7:15_1_1_8"><span class="lab"><font color="Green" title="E36">A33</font></span></a>, <a class="txt" href="#E8:15_1_1_8"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="txt" href="#E9:15_1_1_8"><span class="lab"><font color="Green" title="E38">A35</font></span></a>, <a class="txt" href="#E10:15_1_1_8"><span class="lab"><font color="Green" title="E39">A36</font></span></a>, <a class="ref" href="nat_1.html#T28" title="NAT_1:th.28">NAT_1:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_2"><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="E1:15_1_1_8_2"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:15_1_1_8"><span class="lab"><font color="Green" title="E40">A37</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_8_2_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><a NAME="E2:15_1_1_8_2_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:15_1_1_8_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_8_2_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_2_2"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_8_2_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:15_1_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E24:15_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="topreal1.html#T20" title="TOPREAL1:th.20">TOPREAL1:20</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_8_2_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_8_2_3"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_8_2_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c5">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E26:15_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="topreal1.html#T19" title="TOPREAL1:th.19">TOPREAL1:19</a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">set </span><font color="Maroon" title="c4">Xf1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Xf2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">Xf</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E32:15_1_1"/><span class="lab"><font color="Green" title="E30">A38</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E33:15_1_1"/><span class="lab"><font color="Green" title="E31">A39</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E34:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">Xf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E35:15_1_1"/>
1 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E33">A41</font></span>: <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E37:15_1_1"/>
2 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E34">A42</font></span>: <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E39:15_1_1"/>
3 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A43</font></span>: <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E41:15_1_1"/>
1 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:15_1_1"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E36">A44</font></span>: <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<a NAME="E43:15_1_1"/>
2 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:15_1_1"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E37">A45</font></span>: <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:15_1_1"><span class="lab"><font color="Green" title="E30">A38</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_14"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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> <font color="Maroon" title="c6">Xf</font> implies <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_14"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
<font color="Maroon" title="c7">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> <font color="Maroon" title="c6">Xf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E2:15_1_1_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A47</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:15_1_1_14"/><span class="lab"><font color="Green" title="E40">A48</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:15_1_1"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E1:15_1_1_14"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_14_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="E1:15_1_1_14_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_14"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E3:15_1_1_14"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_14_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_14_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_14_1_1"/>
<font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E36:15_1_1"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_14_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_14_1_2"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_14_1_2"/>
<font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E38:15_1_1"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_14_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_14_1_3"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_14_1_3"/>
<font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E40:15_1_1"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_14_1_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_14_1_4"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_14_1_4"/>
<font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E42:15_1_1"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_14_1_5"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_14_1_5"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_14_1_5"/>
<font color="Maroon" title="c6">Xf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E44:15_1_1"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<a NAME="E46:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A49</font></span>: 
<font color="Maroon" title="c6">Xf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E34:15_1_1"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E39">A50</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T128" title="FINSEQ_2:th.128">FINSEQ_2:128</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T30" title="ENUMSET1:th.30">ENUMSET1:30</a></span>
;<br/>
<a NAME="E48:15_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A51</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:15_1_1"><span class="lab"><font color="Green" title="E39">A50</font></span></a>, <a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>

;<br/>
<a NAME="E50:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A52</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>;<br/>
<span class="lab"><font color="Green" title="E42">A53</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Xf</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:15_1_1"><span class="lab"><font color="Green" title="E40">A51</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>
;<br/>
<a NAME="E52:15_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font>, <font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>) or  not <font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>) or <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c8">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c7">n</font>) )</span><br/>

<a NAME="E1:15_1_1_18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E2:15_1_1_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A54</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:15_1_1_18"/><span class="lab"><font color="Green" title="E44">A55</font></span>: 
<font color="Maroon" title="c7">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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:15_1_1_18"/><span class="lab"><font color="Green" title="E45">A56</font></span>: 
<font color="Maroon" title="c8">m</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c8">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c7">n</font>) )</span><br/>

<a NAME="E5:15_1_1_18"/><span class="lab"><font color="Green" title="E46">A57</font></span>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_18"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="txt" href="#E3:15_1_1_18"><span class="lab"><font color="Green" title="E44">A55</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:15_1_1_18"/><span class="lab"><font color="Green" title="E47">A58</font></span>: 
( <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_18"><span class="lab"><font color="Green" title="E43">A54</font></span></a>, <a class="txt" href="#E4:15_1_1_18"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:15_1_1_18"/><span class="lab"><font color="Green" title="E48">A59</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c8">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c7">n</font>)</span><br/>

<a NAME="E8:15_1_1_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A60</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_18"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="txt" href="#E6:15_1_1_18"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E9:15_1_1_18"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_18"><span class="lab"><font color="Green" title="E46">A57</font></span></a>, <a class="txt" href="#E6:15_1_1_18"><span class="lab"><font color="Green" title="E47">A58</font></span></a>, <a class="txt" href="#E7:15_1_1_18"><span class="lab"><font color="Green" title="E48">A59</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:15_1_1_18"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1_18"><span class="lab"><font color="Green" title="E49">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E53:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A61</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:15_1_1"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E50:15_1_1"><span class="lab"><font color="Green" title="E41">A52</font></span></a>, <a class="txt" href="#E51:15_1_1"><span class="lab"><font color="Green" title="E42">A53</font></span></a>, <a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">Yf1</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Yf2</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Yf</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E55:15_1_1"/><span class="lab"><font color="Green" title="E44">A62</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>;<br/>
<a NAME="E56:15_1_1"/><span class="lab"><font color="Green" title="E45">A63</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E57:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A64</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">Yf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>;<br/>
<a NAME="E58:15_1_1"/>
1 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E47">A65</font></span>: <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E60:15_1_1"/>
2 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E48">A66</font></span>: <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E62:15_1_1"/>
3 <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"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A67</font></span>: <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></span>
;<br/>
<a NAME="E64:15_1_1"/>
1 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:15_1_1"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A68</font></span>: <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<a NAME="E66:15_1_1"/>
2 <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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:15_1_1"><span class="lab"><font color="Green" title="E45">A63</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E51">A69</font></span>: <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:15_1_1"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_24"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">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> <font color="Maroon" title="c9">Yf</font> implies <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:15_1_1_24"/><span class="lab"><font color="Green" title="E52">A70</font></span>: 
<font color="Maroon" title="c10">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> <font color="Maroon" title="c9">Yf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E2:15_1_1_24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A71</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:15_1_1_24"/><span class="lab"><font color="Green" title="E54">A72</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:15_1_1"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="txt" href="#E1:15_1_1_24"><span class="lab"><font color="Green" title="E52">A70</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_24_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="E1:15_1_1_24_1"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_24"><span class="lab"><font color="Green" title="E53">A71</font></span></a>, <a class="txt" href="#E3:15_1_1_24"><span class="lab"><font color="Green" title="E54">A72</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_24_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_24_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_24_1_1"/>
<font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E59:15_1_1"><span class="lab"><font color="Green" title="E47">A65</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_24_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_24_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_24_1_2"/>
<font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E61:15_1_1"><span class="lab"><font color="Green" title="E48">A66</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_24_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_24_1_3"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_24_1_3"/>
<font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E63:15_1_1"><span class="lab"><font color="Green" title="E49">A67</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_24_1_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_24_1_4"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_24_1_4"/>
<font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E65:15_1_1"><span class="lab"><font color="Green" title="E50">A68</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_24_1_5"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_24_1_5"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:15_1_1_24_1_5"/>
<font color="Maroon" title="c9">Yf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E67:15_1_1"><span class="lab"><font color="Green" title="E51">A69</font></span></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<a NAME="E69:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A73</font></span>: 
<font color="Maroon" title="c9">Yf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E57:15_1_1"><span class="lab"><font color="Green" title="E46">A64</font></span></a>, <a class="ref" href="goboard1.html#D2" title="GOBOARD1:def.2">GOBOARD1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A74</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T128" title="FINSEQ_2:th.128">FINSEQ_2:128</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T59" title="ENUMSET1:th.59">ENUMSET1:59</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T30" title="ENUMSET1:th.30">ENUMSET1:30</a></span>
;<br/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>
<br/>.= 
<span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" title="ENUMSET1:th.29">ENUMSET1:29</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E54">A75</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font> = 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:15_1_1"><span class="lab"><font color="Green" title="E53">A74</font></span></a>, <a class="ref" href="finseq_1.html#T31" title="FINSEQ_1:th.31">FINSEQ_1:31</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T2" title="ENUMSET1:th.2">ENUMSET1:2</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T30" title="ENUMSET1:th.30">ENUMSET1:30</a></span>
;<br/>
<a NAME="E73:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A76</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T127" title="FINSEQ_2:th.127">FINSEQ_2:127</a></span>;<br/>
<span class="lab"><font color="Green" title="E56">A77</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> = 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Yf</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:15_1_1"><span class="lab"><font color="Green" title="E54">A75</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>
;<br/>
<a NAME="E75:15_1_1"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_29"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>) or  not <font color="Maroon" title="c11">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>) or <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c11">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c10">n</font>) )</span><br/>

<a NAME="E1:15_1_1_29"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E2:15_1_1_29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A78</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:15_1_1_29"/><span class="lab"><font color="Green" title="E58">A79</font></span>: 
<font color="Maroon" title="c10">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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:15_1_1_29"/><span class="lab"><font color="Green" title="E59">A80</font></span>: 
<font color="Maroon" title="c11">m</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"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c11">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c10">n</font>) )</span><br/>

<a NAME="E5:15_1_1_29"/><span class="lab"><font color="Green" title="E60">A81</font></span>: 
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_29"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E3:15_1_1_29"><span class="lab"><font color="Green" title="E58">A79</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:15_1_1_29"/><span class="lab"><font color="Green" title="E61">A82</font></span>: 
( <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_29"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E4:15_1_1_29"><span class="lab"><font color="Green" title="E59">A80</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:15_1_1_29"/><span class="lab"><font color="Green" title="E62">A83</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c11">m</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K422</a>(<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>,<font color="Maroon" title="c10">n</font>)</span><br/>

<a NAME="E8:15_1_1_29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A84</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_29"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="txt" href="#E6:15_1_1_29"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E9:15_1_1_29"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:15_1_1_29"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="txt" href="#E6:15_1_1_29"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="txt" href="#E7:15_1_1_29"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:15_1_1_29"/>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1_29"><span class="lab"><font color="Green" title="E63">A84</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E76:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A85</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:15_1_1"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="txt" href="#E73:15_1_1"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E74:15_1_1"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="seq_4.html#D21" title="SEQ_4:def.21">SEQ_4:def 21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">M</font> = (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>);<br/>
<span class="lab"><font color="Green" title="E58">A86</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> = 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A87</font></span>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> = 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E76:15_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>
;<br/>
<a NAME="E79:15_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> holds <br/><span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_32"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c12">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> implies <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><br/>

<a NAME="E2:15_1_1_32"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="matrix_2.html#T3" title="MATRIX_2:th.3">MATRIX_2:3</a></span>;<br/>
<a NAME="E3:15_1_1_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A88</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></span>;<br/>
<a NAME="E4:15_1_1_32"/><span class="lab"><font color="Green" title="E61">A89</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E5:15_1_1_32"/><span class="lab"><font color="Green" title="E62">A90</font></span>: 
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_32_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="E1:15_1_1_32_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_32"><span class="lab"><font color="Green" title="E60">A88</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_32_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_32_1_1"/><span class="lab"><font color="Green" title="E63">A91</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><br/><div class="add"><a NAME="E2:15_1_1_32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A92</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E3:15_1_1_32_1_1"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_32_1_1"><span class="lab"><font color="Green" title="E63">A91</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_32_1_1"/>
<span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E76:15_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E4:15_1_1_32"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="txt" href="#E2:15_1_1_32_1_1"><span class="lab"><font color="Green" title="E64">A92</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_32_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_32_1_2"/><span class="lab"><font color="Green" title="E63">A93</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><br/><div class="add"><a NAME="E2:15_1_1_32_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A94</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E3:15_1_1_32_1_2"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_32_1_2"><span class="lab"><font color="Green" title="E63">A93</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_32_1_2"/>
<span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E76:15_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E4:15_1_1_32"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="txt" href="#E5:15_1_1_32"><span class="lab"><font color="Green" title="E62">A90</font></span></a>, <a class="txt" href="#E2:15_1_1_32_1_2"><span class="lab"><font color="Green" title="E64">A94</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_32_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_32_1_3"/><span class="lab"><font color="Green" title="E63">A95</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><br/><div class="add"><a NAME="E2:15_1_1_32_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A96</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E3:15_1_1_32_1_3"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_32_1_3"><span class="lab"><font color="Green" title="E63">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_32_1_3"/>
<span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E76:15_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E4:15_1_1_32"><span class="lab"><font color="Green" title="E61">A89</font></span></a>, <a class="txt" href="#E5:15_1_1_32"><span class="lab"><font color="Green" title="E62">A90</font></span></a>, <a class="txt" href="#E2:15_1_1_32_1_3"><span class="lab"><font color="Green" title="E64">A96</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_32_1_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_32_1_4"/><span class="lab"><font color="Green" title="E63">A97</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span></span><br/><div class="add"><a NAME="E2:15_1_1_32_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A98</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E3:15_1_1_32_1_4"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_32_1_4"><span class="lab"><font color="Green" title="E63">A97</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:15_1_1_32_1_4"/>
<span class="p1">(<span class="default">(<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">n</font>,<font color="Maroon" title="c12">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:15_1_1"><span class="lab"><font color="Green" title="E43">A61</font></span></a>, <a class="txt" href="#E76:15_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E5:15_1_1_32"><span class="lab"><font color="Green" title="E62">A90</font></span></a>, <a class="txt" href="#E2:15_1_1_32_1_4"><span class="lab"><font color="Green" title="E64">A98</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">then </span><span class="lab"><font color="Green" title="E60">A99</font></span>: (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> (<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>,<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default">1,1</span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>) = 
 <a href="goboard2.html#K1" title="GOBOARD2:func.1">GoB</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K28" title="SEQ_4:func.28">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E77:15_1_1"><span class="lab"><font color="Green" title="E58">A86</font></span></a>, <a class="txt" href="#E78:15_1_1"><span class="lab"><font color="Green" title="E59">A87</font></span></a>, <a class="ref" href="goboard2.html#D1" title="GOBOARD2:def.1">GOBOARD2:def 1</a></span>
<br/>.= 
 <a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#D2" title="GOBOARD2:def.2">GOBOARD2:def 2</a></span>
;<br/>
<a NAME="E81:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A100</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>;<br/>
<a NAME="E82:15_1_1"/><span class="lab"><font color="Green" title="E62">A101</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>;<br/>
<a NAME="E83:15_1_1"/><span class="lab"><font color="Green" title="E63">A102</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>;<br/>
<a NAME="E84:15_1_1"/><span class="lab"><font color="Green" title="E64">A103</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E85:15_1_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a>,<a class="ref" href="goboard5.html#D5" target="_self" title="GOBOARD5:def.5">GOBOARD5:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></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> <font color="Maroon" title="c3">f</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> or  for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>) or <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_34"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</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> <font color="Maroon" title="c3">f</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_1_34"/><span class="lab"><font color="Green" title="E65">A104</font></span>: 
<font color="Maroon" title="c11">k</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> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/>

<a NAME="E2:15_1_1_34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A105</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 5
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E3:15_1_1_34"/><span class="lab"><font color="Green" title="E67">A106</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15_1_1_34"><span class="lab"><font color="Green" title="E65">A104</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_34_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="E1:15_1_1_34_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_1_34"><span class="lab"><font color="Green" title="E66">A105</font></span></a>, <a class="txt" href="#E3:15_1_1_34"><span class="lab"><font color="Green" title="E67">A106</font></span></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_34_1_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_34_1_1"/><span class="lab"><font color="Green" title="E68">A107</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><div class="add"><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) )</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_34_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span><br/><span class="kw">thus </span><a NAME="E3:15_1_1_34_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="txt" href="#E1:15_1_1_34_1_1"><span class="lab"><font color="Green" title="E68">A107</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_34_1_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_34_1_2"/><span class="lab"><font color="Green" title="E68">A108</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><div class="add"><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2) )</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_34_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2)</span><br/><span class="kw">thus </span><a NAME="E3:15_1_1_34_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,2)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15_1_1"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="txt" href="#E1:15_1_1_34_1_2"><span class="lab"><font color="Green" title="E68">A108</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_34_1_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_34_1_3"/><span class="lab"><font color="Green" title="E68">A109</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><div class="add"><span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2) )</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_34_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2)</span><br/><span class="kw">thus </span><a NAME="E3:15_1_1_34_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,2)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:15_1_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="txt" href="#E1:15_1_1_34_1_3"><span class="lab"><font color="Green" title="E68">A109</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_34_1_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_34_1_4"/><span class="lab"><font color="Green" title="E68">A110</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><div class="add"><span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1) )</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_34_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1)</span><br/><span class="kw">thus </span><a NAME="E3:15_1_1_34_1_4"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (2,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15_1_1"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="txt" href="#E1:15_1_1_34_1_4"><span class="lab"><font color="Green" title="E68">A110</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_34_1_5"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_34_1_5"/><span class="lab"><font color="Green" title="E68">A111</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><div class="add"><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) )</span><br/><span class="kw">thus </span><a NAME="E2:15_1_1_34_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span><br/><span class="kw">thus </span><a NAME="E3:15_1_1_34_1_5"/>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="txt" href="#E1:15_1_1_34_1_5"><span class="lab"><font color="Green" title="E68">A111</font></span></a>, <a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">let </span><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">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> <font color="Maroon" title="c3">f</font> or  not <font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> or  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) or <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E86:15_1_1"/><span class="lab"><font color="Green" title="E65">A112</font></span>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E87:15_1_1"/><span class="lab"><font color="Green" title="E66">A113</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) or <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c12">m</font>, <font color="Maroon" title="c13">k</font>, <font color="Maroon" title="c14">i</font>, <font color="Maroon" title="c15">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font>) or  not <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>) or <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E88:15_1_1"/><span class="lab"><font color="Green" title="E67">A114</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E89:15_1_1"/><span class="lab"><font color="Green" title="E68">A115</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E90:15_1_1"/><span class="lab"><font color="Green" title="E69">A116</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">m</font>,<font color="Maroon" title="c13">k</font>)
 <span class="kw">and </span><br/><a NAME="E91:15_1_1"/><span class="lab"><font color="Green" title="E70">A117</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E92:15_1_1"/><span class="lab"><font color="Green" title="E71">A118</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E86:15_1_1"><span class="lab"><font color="Green" title="E65">A112</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E93:15_1_1"/>
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E87:15_1_1"><span class="lab"><font color="Green" title="E66">A113</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E94:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A119</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_35"><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="E1:15_1_1_35"/>
( <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E92:15_1_1"><span class="lab"><font color="Green" title="E71">A118</font></span></a>, <a class="txt" href="#E94:15_1_1"><span class="lab"><font color="Green" title="E72">A119</font></span></a>, <a class="ref" href="nat_1.html#T28" title="NAT_1:th.28">NAT_1:28</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_35_1"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_35_1"/><span class="lab"><font color="Green" title="E73">A120</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:15_1_1_35_1"/><span class="lab"><font color="Green" title="E74">A121</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E3:15_1_1_35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A122</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:15_1_1"><span class="lab"><font color="Green" title="E61">A100</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_1"><span class="lab"><font color="Green" title="E73">A120</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:15_1_1_35_1"/><span class="lab"><font color="Green" title="E76">A123</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E81:15_1_1"><span class="lab"><font color="Green" title="E61">A100</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_1"><span class="lab"><font color="Green" title="E73">A120</font></span></a>, <a class="txt" href="#E2:15_1_1_35_1"><span class="lab"><font color="Green" title="E74">A121</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:15_1_1_35_1"/><span class="lab"><font color="Green" title="E77">A124</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E6:15_1_1_35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A125</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15_1_1"><span class="lab"><font color="Green" title="E62">A101</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_1"><span class="lab"><font color="Green" title="E73">A120</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:15_1_1_35_1"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15_1_1"><span class="lab"><font color="Green" title="E62">A101</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_1"><span class="lab"><font color="Green" title="E73">A120</font></span></a>, <a class="txt" href="#E5:15_1_1_35_1"><span class="lab"><font color="Green" title="E77">A124</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E8:15_1_1_35_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_1_35_1"><span class="lab"><font color="Green" title="E76">A123</font></span></a>, <a class="ref" href="seqm_3.html#T41" title="SEQM_3:th.41">SEQM_3:41</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_1_1_35_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_35_1"><span class="lab"><font color="Green" title="E75">A122</font></span></a>, <a class="txt" href="#E6:15_1_1_35_1"><span class="lab"><font color="Green" title="E78">A125</font></span></a>, <a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_35_2"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_35_2"/><span class="lab"><font color="Green" title="E73">A126</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:15_1_1_35_2"/><span class="lab"><font color="Green" title="E74">A127</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E3:15_1_1_35_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A128</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15_1_1"><span class="lab"><font color="Green" title="E62">A101</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_2"><span class="lab"><font color="Green" title="E73">A126</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:15_1_1_35_2"/><span class="lab"><font color="Green" title="E76">A129</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E82:15_1_1"><span class="lab"><font color="Green" title="E62">A101</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_2"><span class="lab"><font color="Green" title="E73">A126</font></span></a>, <a class="txt" href="#E2:15_1_1_35_2"><span class="lab"><font color="Green" title="E74">A127</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:15_1_1_35_2"/><span class="lab"><font color="Green" title="E77">A130</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E6:15_1_1_35_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A131</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:15_1_1"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_2"><span class="lab"><font color="Green" title="E73">A126</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:15_1_1_35_2"/><span class="lab"><font color="Green" title="E79">A132</font></span>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:15_1_1"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_2"><span class="lab"><font color="Green" title="E73">A126</font></span></a>, <a class="txt" href="#E5:15_1_1_35_2"><span class="lab"><font color="Green" title="E77">A130</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E8:15_1_1_35_2"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_35_2"><span class="lab"><font color="Green" title="E75">A128</font></span></a>, <a class="txt" href="#E6:15_1_1_35_2"><span class="lab"><font color="Green" title="E78">A131</font></span></a>, <a class="ref" href="seqm_3.html#T41" title="SEQM_3:th.41">SEQM_3:41</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_1_1_35_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_1_35_2"><span class="lab"><font color="Green" title="E76">A129</font></span></a>, <a class="txt" href="#E7:15_1_1_35_2"><span class="lab"><font color="Green" title="E79">A132</font></span></a>, <a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_35_3"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_35_3"/><span class="lab"><font color="Green" title="E73">A133</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:15_1_1_35_3"/><span class="lab"><font color="Green" title="E74">A134</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E3:15_1_1_35_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A135</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:15_1_1"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_3"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:15_1_1_35_3"/><span class="lab"><font color="Green" title="E76">A136</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:15_1_1"><span class="lab"><font color="Green" title="E63">A102</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_3"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="txt" href="#E2:15_1_1_35_3"><span class="lab"><font color="Green" title="E74">A134</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:15_1_1_35_3"/><span class="lab"><font color="Green" title="E77">A137</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E6:15_1_1_35_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A138</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:15_1_1"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_3"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:15_1_1_35_3"/>
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:15_1_1"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_3"><span class="lab"><font color="Green" title="E73">A133</font></span></a>, <a class="txt" href="#E5:15_1_1_35_3"><span class="lab"><font color="Green" title="E77">A137</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E8:15_1_1_35_3"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_1_35_3"><span class="lab"><font color="Green" title="E76">A136</font></span></a>, <a class="ref" href="seqm_3.html#T41" title="SEQM_3:th.41">SEQM_3:41</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_1_1_35_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_35_3"><span class="lab"><font color="Green" title="E75">A135</font></span></a>, <a class="txt" href="#E6:15_1_1_35_3"><span class="lab"><font color="Green" title="E78">A138</font></span></a>, <a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="15_1_1_35_4"><span class="kw">suppose </span></a><a NAME="E1:15_1_1_35_4"/><span class="lab"><font color="Green" title="E73">A139</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><div class="add"><a NAME="E2:15_1_1_35_4"/><span class="lab"><font color="Green" title="E74">A140</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E3:15_1_1_35_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A141</font></span>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:15_1_1"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_4"><span class="lab"><font color="Green" title="E73">A139</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E4:15_1_1_35_4"/><span class="lab"><font color="Green" title="E76">A142</font></span>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:15_1_1"><span class="lab"><font color="Green" title="E64">A103</font></span></a>, <a class="txt" href="#E88:15_1_1"><span class="lab"><font color="Green" title="E67">A114</font></span></a>, <a class="txt" href="#E90:15_1_1"><span class="lab"><font color="Green" title="E69">A116</font></span></a>, <a class="txt" href="#E1:15_1_1_35_4"><span class="lab"><font color="Green" title="E73">A139</font></span></a>, <a class="txt" href="#E2:15_1_1_35_4"><span class="lab"><font color="Green" title="E74">A140</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:15_1_1_35_4"/><span class="lab"><font color="Green" title="E77">A143</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E80:15_1_1"><span class="lab"><font color="Green" title="E60">A99</font></span></a>, <a class="ref" href="matrix_2.html#T4" title="MATRIX_2:th.4">MATRIX_2:4</a></span>;<br/><a NAME="E6:15_1_1_35_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A144</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E81:15_1_1"><span class="lab"><font color="Green" title="E61">A100</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_4"><span class="lab"><font color="Green" title="E73">A139</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:15_1_1_35_4"/><span class="lab"><font color="Green" title="E79">A145</font></span>: 
<font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:15_1_1"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E81:15_1_1"><span class="lab"><font color="Green" title="E61">A100</font></span></a>, <a class="txt" href="#E89:15_1_1"><span class="lab"><font color="Green" title="E68">A115</font></span></a>, <a class="txt" href="#E91:15_1_1"><span class="lab"><font color="Green" title="E70">A117</font></span></a>, <a class="txt" href="#E1:15_1_1_35_4"><span class="lab"><font color="Green" title="E73">A139</font></span></a>, <a class="txt" href="#E5:15_1_1_35_4"><span class="lab"><font color="Green" title="E77">A143</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E8:15_1_1_35_4"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1_35_4"><span class="lab"><font color="Green" title="E75">A141</font></span></a>, <a class="txt" href="#E6:15_1_1_35_4"><span class="lab"><font color="Green" title="E78">A144</font></span></a>, <a class="ref" href="seqm_3.html#T41" title="SEQM_3:th.41">SEQM_3:41</a></span>;<br/><span class="kw">hence </span><a NAME="E9:15_1_1_35_4"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15_1_1_35_4"><span class="lab"><font color="Green" title="E76">A142</font></span></a>, <a class="txt" href="#E7:15_1_1_35_4"><span class="lab"><font color="Green" title="E79">A145</font></span></a>, <a class="ref" href="seqm_3.html#T42" title="SEQM_3:th.42">SEQM_3:42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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