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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">G1</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a><br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being  <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>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font> &amp; $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="Olive" title="b2">g1</font> &amp; 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="Olive" title="b3">g2</font> &amp; <font color="Olive" title="b2">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b3">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b2">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">g1</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="Olive" title="b2">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b3">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b3">g2</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="Olive" title="b3">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <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>  st 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="Olive" title="b1">f1</font> &amp; 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="Olive" title="b2">f2</font> &amp; <font color="Olive" title="b1">f1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b2">f2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Olive" title="b1">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Olive" title="b1">f1</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="Olive" title="b1">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Olive" title="b2">f2</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="Olive" title="b2">f2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">f2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f1</font>, <font color="Maroon" title="c3">f2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f1</font> &amp; 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">f2</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c2">f1</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="c2">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c3">f2</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">f2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f2</font> )</span><br/>





<a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">G1</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <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>  st <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="Olive" title="b1">g1</font> &amp; 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="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b1">g1</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="Olive" title="b1">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">g2</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="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="c6">g1</font> &amp; 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="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c6">g1</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="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c7">g2</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="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1"/>
<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:1_1"/><span class="lab"><font color="Green" title="E5">A4</font></span>: 
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="c6">g1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
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="c7">g2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1"/><span class="lab"><font color="Green" title="E7">A6</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1"/><span class="lab"><font color="Green" title="E8">A7</font></span>: 
<font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1"/><span class="lab"><font color="Green" title="E9">A8</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:1_1"/><span class="lab"><font color="Green" title="E10">A9</font></span>: 
<font color="Maroon" title="c6">g1</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="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:1_1"/><span class="lab"><font color="Green" title="E11">A10</font></span>: 
<font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:1_1"/><span class="lab"><font color="Green" title="E12">A11</font></span>: 
<font color="Maroon" title="c7">g2</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="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> );<br/>
<a NAME="E12:1_1"/><span class="lab"><font color="Green" title="E13">A12</font></span>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c8">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">n</font>]</span><br/>

<span class="kw">thus </span><a NAME="E1:1_1_1"/>
<font color="Maroon" title="c8">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="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E2:1_1_1"/>
<font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1"><span class="lab"><font color="Green" title="E12">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:1_1"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">m</font>] &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E12:1_1"><span class="lab"><font color="Green" title="E13">A12</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g</font> = <font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font>;<br/>
<a NAME="E16:1_1"/><span class="lab"><font color="Green" title="E15">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></span>;<br/>
<a NAME="E17:1_1"/><span class="lab"><font color="Green" title="E16">A15</font></span>: 
<font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></span>;<br/>
<a NAME="E18:1_1"/><span class="lab"><font color="Green" title="E17">A16</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E19:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A17</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></span>;<br/>
<a NAME="E20:1_1"/><span class="lab"><font color="Green" title="E19">A18</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">n</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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_2"/><span class="lab"><font color="Green" title="E20">A19</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_2"/><span class="lab"><font color="Green" title="E21">A20</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E5:1_1_2"/><span class="lab"><font color="Green" title="E22">A21</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E3:1_1_2"><span class="lab"><font color="Green" title="E20">A19</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<font color="Maroon" title="c12">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="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<a NAME="E7:1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E4:1_1_2"><span class="lab"><font color="Green" title="E21">A20</font></span></a>, <a class="txt" href="#E5:1_1_2"><span class="lab"><font color="Green" title="E22">A21</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_2"><span class="lab"><font color="Green" title="E23">A22</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c11">L1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1, <font color="Maroon" title="c12">Ll</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span> as   <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/>
<a NAME="E22:1_1"/><span class="lab"><font color="Green" title="E20">A23</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E23:1_1"/><span class="lab"><font color="Green" title="E21">A24</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E24:1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/>
<a NAME="E25:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A25</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E26:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A26</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> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E27:1_1"/><span class="lab"><font color="Green" title="E24">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1"><span class="lab"><font color="Green" title="E22">A25</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E28:1_1"/><span class="lab"><font color="Green" title="E25">A28</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="goboard1.html#T23" title="GOBOARD1:th.23">GOBOARD1:23</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,$1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/>
<a NAME="E29:1_1"/><span class="lab"><font color="Green" title="E26">A29</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E30:1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/>
<a NAME="E31:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A30</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<a NAME="E32:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A31</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E33:1_1"/><span class="lab"><font color="Green" title="E29">A32</font></span>: 
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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></span>;<br/>
<a NAME="E34:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A33</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">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E35:1_1"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c13">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:1_1_3"/><span class="lab"><font color="Green" title="E32">A35</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_3"/><span class="lab"><font color="Green" title="E33">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c13">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">i</font>]</span><br/>

<a NAME="E5:1_1_3"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_3"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_3"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E8:1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A37</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</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="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E3:1_1_3"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/>
<a NAME="E9:1_1_3"/><span class="lab"><font color="Green" title="E35">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E10:1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E11:1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E12:1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E8:1_1_3"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:1_1_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_3"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c13">mi</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E37:1_1"/><span class="lab"><font color="Green" title="E32">A39</font></span>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c13">mi</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c13">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E35:1_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a>);<br/></span>
<a NAME="E38:1_1"/><span class="lab"><font color="Green" title="E33">A40</font></span>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:1_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">ma</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E40:1_1"/><span class="lab"><font color="Green" title="E34">A41</font></span>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c14">ma</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ma</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E29:1_1"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="#E38:1_1"><span class="lab"><font color="Green" title="E33">A40</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c15">mi</font> = <font color="Maroon" title="c13">mi</font>, <font color="Maroon" title="c16">ma</font> = <font color="Maroon" title="c14">ma</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E42:1_1"/><span class="lab"><font color="Green" title="E35">A42</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">Lmi</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font> as   <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">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> );<br/>
<a NAME="E44:1_1"/><span class="lab"><font color="Green" title="E36">A43</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E45:1_1"/><span class="lab"><font color="Green" title="E37">A44</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E46:1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E8:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="txt" href="#E9:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E42:1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></span>;<br/>
<a NAME="E47:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A45</font></span>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">ma1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E49:1_1"/><span class="lab"><font color="Green" title="E39">A46</font></span>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c18">ma1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">ma1</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E44:1_1"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E47:1_1"><span class="lab"><font color="Green" title="E38">A45</font></span></a>);<br/></span>
<a NAME="E50:1_1"/><span class="lab"><font color="Green" title="E40">A47</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E51:1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">l1</font> = <font color="Maroon" title="c15">mi</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1, <font color="Maroon" title="c20">l2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:1_1"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/>
<a NAME="E53:1_1"/><span class="lab"><font color="Green" title="E41">A48</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E54:1_1"/><span class="lab"><font color="Green" title="E42">A49</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">ma1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">ma1</font> = <font color="Maroon" title="c18">ma1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c22">k1</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> <span class="kw"> such that </span><br/><a NAME="E57:1_1"/><span class="lab"><font color="Green" title="E43">A50</font></span>: 
<font color="Maroon" title="c22">k1</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="c17">Lmi</font>
 <span class="kw">and </span><br/><a NAME="E58:1_1"/><span class="lab"><font color="Green" title="E44">A51</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E59:1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">Lmi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">Lmi</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E60:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A52</font></span>: 
<font color="Maroon" title="c22">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E61:1_1"/><span class="lab"><font color="Green" title="E46">A53</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E62:1_1"/><span class="lab"><font color="Green" title="E47">A54</font></span>: 
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c23">n</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> <span class="kw"> such that </span><br/><a NAME="E2:1_1_4"/><span class="lab"><font color="Green" title="E48">A55</font></span>: 
<font color="Maroon" title="c23">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">g1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_4"/><span class="lab"><font color="Green" title="E49">A56</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E8:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="txt" href="#E9:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E42:1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="txt" href="#E45:1_1"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c24">i</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_4"/><span class="lab"><font color="Green" title="E50">A57</font></span>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_4"/><span class="lab"><font color="Green" title="E51">A58</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_4"><span class="lab"><font color="Green" title="E49">A56</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E7:1_1_4"/><span class="lab"><font color="Green" title="E52">A59</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:1_1_4"/><span class="lab"><font color="Green" title="E53">A60</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E9:1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c24">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">m</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> <span class="kw"> such that </span><br/><a NAME="E11:1_1_4"/><span class="lab"><font color="Green" title="E54">A61</font></span>: 
<font color="Maroon" title="c25">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">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E12:1_1_4"/><span class="lab"><font color="Green" title="E55">A62</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c24">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E16:1_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E7:1_1_4"><span class="lab"><font color="Green" title="E52">A59</font></span></a>, <a class="ref" href="goboard1.html#T49" title="GOBOARD1:th.49">GOBOARD1:49</a></span>;<br/>
<a NAME="E13:1_1_4"/><span class="lab"><font color="Green" title="E56">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_4"><span class="lab"><font color="Green" title="E54">A61</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">Ci</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c24">i</font> as   <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/>
<a NAME="E15:1_1_4"/><span class="lab"><font color="Green" title="E57">A64</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E58">A65</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c24">i</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><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><a NAME="E17:1_1_4"/><span class="lab"><font color="Green" title="E59">A66</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E18:1_1_4"/><span class="lab"><font color="Green" title="E60">A67</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">j</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> <span class="kw"> such that </span><br/><a NAME="E20:1_1_4"/><span class="lab"><font color="Green" title="E61">A68</font></span>: 
<font color="Maroon" title="c27">j</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="c26">Ci</font>
 <span class="kw">and </span><br/><a NAME="E21:1_1_4"/><span class="lab"><font color="Green" title="E62">A69</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Ci</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_4"><span class="lab"><font color="Green" title="E55">A62</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">Lj</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c27">j</font> as   <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/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">Lj</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E24:1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A70</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">Lj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_4"><span class="lab"><font color="Green" title="E60">A67</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A71</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font> = 
<font color="Maroon" title="c26">Ci</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_4"><span class="lab"><font color="Green" title="E61">A68</font></span></a>, <a class="txt" href="#E21:1_1_4"><span class="lab"><font color="Green" title="E62">A69</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c28">Lj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E18:1_1_4"><span class="lab"><font color="Green" title="E60">A67</font></span></a>, <a class="txt" href="#E8:1_1_4"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="txt" href="#E16:1_1_4"><span class="lab"><font color="Green" title="E58">A65</font></span></a>, <a class="txt" href="#E20:1_1_4"><span class="lab"><font color="Green" title="E61">A68</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>
<br/>.= 
<font color="Maroon" title="c28">Lj</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E24:1_1_4"><span class="lab"><font color="Green" title="E63">A70</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<a NAME="E26:1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c28">Lj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E27:1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c28">Lj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_4"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="txt" href="#E18:1_1_4"><span class="lab"><font color="Green" title="E60">A67</font></span></a>, <a class="txt" href="#E8:1_1_4"><span class="lab"><font color="Green" title="E53">A60</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E28:1_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c28">Lj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_4"><span class="lab"><font color="Green" title="E64">A71</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E29:1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A72</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Ci</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Ci</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c27">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_4"><span class="lab"><font color="Green" title="E56">A63</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A73</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_4_4"/>
<font color="Maroon" title="c27">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">mi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_4_4"/><span class="kw">then </span>
( <font color="Maroon" title="c27">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mi</font> or <font color="Maroon" title="c27">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">mi</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_4_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E61:1_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E17:1_1_4"><span class="lab"><font color="Green" title="E59">A66</font></span></a>, <a class="txt" href="#E15:1_1_4"><span class="lab"><font color="Green" title="E57">A64</font></span></a>, <a class="txt" href="#E20:1_1_4"><span class="lab"><font color="Green" title="E61">A68</font></span></a>, <a class="txt" href="#E29:1_1_4"><span class="lab"><font color="Green" title="E65">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E31:1_1_4"/>
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_4"><span class="lab"><font color="Green" title="E48">A55</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:1_1_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E6:1_1_4"><span class="lab"><font color="Green" title="E51">A58</font></span></a>, <a class="txt" href="#E13:1_1_4"><span class="lab"><font color="Green" title="E56">A63</font></span></a>, <a class="txt" href="#E25:1_1_4"><span class="lab"><font color="Green" title="E64">A71</font></span></a>, <a class="txt" href="#E30:1_1_4"><span class="lab"><font color="Green" title="E66">A73</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E63:1_1"/><span class="lab"><font color="Green" title="E48">A74</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> $1,<font color="Maroon" title="c22">k1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Ck1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c22">k1</font> as   <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">consider </span><font color="Maroon" title="c24">h1</font> being   <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><span class="kw"> such that </span><br/><a NAME="E66:1_1"/><span class="lab"><font color="Green" title="E49">A75</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l1</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">h1</font> holds <br/><font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" title="FINSEQ_4:sch.2">FINSEQ_4:sch 2</a>();<br/></span>
<a NAME="E67:1_1"/><span class="lab"><font color="Green" title="E50">A76</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_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="E1:1_1_5_1"/>
( <font color="Maroon" title="c4">k</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">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_1"/><span class="lab"><font color="Green" title="E51">A77</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">c1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1 as   <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">consider </span><font color="Maroon" title="c26">x</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_1"/><span class="lab"><font color="Green" title="E52">A78</font></span>: 
<font color="Maroon" title="c26">x</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="c25">c1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_1"/><span class="lab"><font color="Green" title="E53">A79</font></span>: 
<font color="Maroon" title="c7">g2</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="c25">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">lx</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c26">x</font> as   <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/><a NAME="E7:1_1_5_1_1"/><span class="lab"><font color="Green" title="E54">A80</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_1"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:1_1_5_1_1"/><span class="lab"><font color="Green" title="E55">A81</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E9:1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_1"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">m</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> <span class="kw"> such that </span><br/><a NAME="E11:1_1_5_1_1"/><span class="lab"><font color="Green" title="E56">A82</font></span>: 
<font color="Maroon" title="c28">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> <font color="Maroon" title="c6">g1</font>
 <span class="kw">and </span><br/><a NAME="E12:1_1_5_1_1"/><span class="lab"><font color="Green" title="E57">A83</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c27">lx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E8:1_1"><span class="lab"><font color="Green" title="E9">A8</font></span></a>, <a class="txt" href="#E9:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E7:1_1_5_1_1"><span class="lab"><font color="Green" title="E54">A80</font></span></a>, <a class="ref" href="goboard1.html#T45" title="GOBOARD1:th.45">GOBOARD1:45</a></span>;<br/><a NAME="E13:1_1_5_1_1"/><span class="lab"><font color="Green" title="E58">A84</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_1"><span class="lab"><font color="Green" title="E56">A82</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E14:1_1_5_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E15:1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A85</font></span>: 
<font color="Maroon" title="c26">x</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_1"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="txt" href="#E8:1_1_5_1_1"><span class="lab"><font color="Green" title="E55">A81</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E16:1_1_5_1_1"/><span class="lab"><font color="Green" title="E60">A86</font></span>: 
<font color="Maroon" title="c25">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_1"><span class="lab"><font color="Green" title="E52">A78</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c29">y</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> <span class="kw"> such that </span><br/><a NAME="E18:1_1_5_1_1"/><span class="lab"><font color="Green" title="E61">A87</font></span>: 
<font color="Maroon" title="c29">y</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="c27">lx</font>
 <span class="kw">and </span><br/><a NAME="E19:1_1_5_1_1"/><span class="lab"><font color="Green" title="E62">A88</font></span>: 
<font color="Maroon" title="c27">lx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_1"><span class="lab"><font color="Green" title="E57">A83</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E20:1_1_5_1_1"/>
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">lx</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E21:1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A89</font></span>: 
<font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E1:1_1_5_1_1"><span class="lab"><font color="Green" title="E51">A77</font></span></a>, <a class="txt" href="#E18:1_1_5_1_1"><span class="lab"><font color="Green" title="E61">A87</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E22:1_1_5_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/><a NAME="E23:1_1_5_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E24:1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A90</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E25:1_1_5_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> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E26:1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A91</font></span>: 
<font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E27:1_1_5_1_1"/>
<font color="Maroon" title="c27">lx</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_1"><span class="lab"><font color="Green" title="E61">A87</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E28:1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_1"><span class="lab"><font color="Green" title="E53">A79</font></span></a>, <a class="txt" href="#E24:1_1_5_1_1"><span class="lab"><font color="Green" title="E64">A90</font></span></a>, <a class="txt" href="#E15:1_1_5_1_1"><span class="lab"><font color="Green" title="E59">A85</font></span></a>, <a class="txt" href="#E19:1_1_5_1_1"><span class="lab"><font color="Green" title="E62">A88</font></span></a>, <a class="txt" href="#E16:1_1_5_1_1"><span class="lab"><font color="Green" title="E60">A86</font></span></a>, <a class="txt" href="#E21:1_1_5_1_1"><span class="lab"><font color="Green" title="E63">A89</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><span class="kw">hence </span><a NAME="E29:1_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_1"><span class="lab"><font color="Green" title="E58">A84</font></span></a>, <a class="txt" href="#E26:1_1_5_1_1"><span class="lab"><font color="Green" title="E65">A91</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2"/><span class="lab"><font color="Green" title="E51">A92</font></span>: 
<font color="Maroon" title="c4">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A93</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_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="E1:1_1_5_1_2_1_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> or <font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_5_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:1_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A94</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">ma1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">w</font> = <font color="Maroon" title="c21">ma1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">Lma</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">ma</font> as   <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/><a NAME="E5:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E54">A95</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</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="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E55">A96</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="c24">h1</font> implies <font color="Maroon" title="c27">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="c5">G1</font> )</span><br/><a NAME="E1:1_1_5_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E56">A97</font></span>: 
<font color="Maroon" title="c19">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></span>;<br/><span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A98</font></span>: 
<font color="Maroon" title="c27">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="c24">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">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="c5">G1</font></span><br/><a NAME="E3:1_1_5_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A99</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_1"/>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E57">A98</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A100</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E56">A97</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_1"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E59">A100</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">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_1"/>
<font color="Maroon" title="c27">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E58">A99</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E56">A101</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">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="c24">h1</font> &amp; <font color="Maroon" title="c27">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="c24">h1</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E57">A102</font></span>: 
<font color="Maroon" title="c27">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="c24">h1</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E58">A103</font></span>: 
<font color="Maroon" title="c27">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="c24">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c27">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E58">A103</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A104</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">k1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c28">i1</font>, <font color="Maroon" title="c29">i2</font>, <font color="Maroon" title="c30">j1</font>, <font color="Maroon" title="c31">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">i1</font>,<font color="Maroon" title="c29">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">j1</font>,<font color="Maroon" title="c31">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">i1</font>,<font color="Maroon" title="c29">i2</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c30">j1</font>,<font color="Maroon" title="c31">j2</font> implies <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="c28">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c30">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c29">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">j2</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="E5:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E60">A105</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">i1</font>,<font color="Maroon" title="c29">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E61">A106</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">j1</font>,<font color="Maroon" title="c31">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E62">A107</font></span>: 
<font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">i1</font>,<font color="Maroon" title="c29">i2</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E63">A108</font></span>: 
<font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c30">j1</font>,<font color="Maroon" title="c31">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c30">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c29">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E9:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E58">A103</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A109</font></span>: 
( <font color="Maroon" title="c30">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c31">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E59">A104</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E61">A106</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E63">A108</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c27">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E57">A102</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A110</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c22">k1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E57">A102</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c28">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font> &amp; <font color="Maroon" title="c29">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E65">A110</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E60">A105</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E62">A107</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c28">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c30">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c29">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c31">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c27">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E64">A109</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A111</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c27">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E58">A112</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">n1</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E59">A113</font></span>: 
<font color="Maroon" title="c28">n1</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="c24">h1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E60">A114</font></span>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E61">A115</font></span>: 
<font color="Maroon" title="c28">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E59">A113</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c28">n1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E59">A113</font></span></a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A116</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">n1</font>,<font color="Maroon" title="c22">k1</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> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E58">A112</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">n2</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> <span class="kw"> such that </span><br/><a NAME="E12:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E63">A117</font></span>: 
<font color="Maroon" title="c29">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E64">A118</font></span>: 
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E65">A119</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E63">A117</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c30">i1</font>, <font color="Maroon" title="c31">i2</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> <span class="kw"> such that </span><br/><a NAME="E16:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E66">A120</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">i1</font>,<font color="Maroon" title="c31">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E17:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E67">A121</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c30">i1</font>,<font color="Maroon" title="c31">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E63">A117</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c30">i1</font> as   <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/>
<a NAME="E19:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E68">A122</font></span>: 
<font color="Maroon" title="c31">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E66">A120</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E69">A123</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c32">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c32">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c32">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E68">A122</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E67">A121</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E68">A122</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E68">A122</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E69">A123</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E24:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A124</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c32">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E65">A119</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E25:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c30">i1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E66">A120</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E26:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A125</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E70">A124</font></span></a></span>;<br/>
<a NAME="E27:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon" title="c27">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c28">n1</font>,<font color="Maroon" title="c22">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E59">A113</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E60">A114</font></span></a></span>;<br/>
<a NAME="E28:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c30">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E64">A118</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E66">A120</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E67">A121</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E62">A116</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E29:1_1_5_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E61">A115</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E71">A125</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">hence </span><a NAME="E30:1_1_5_1_2_1_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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> <font color="Maroon" title="c6">g1</font> &amp; <font color="Maroon" title="c21">ma1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">ma</font></span>)</span> );<br/><a NAME="E9:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E58">A126</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A127</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A94</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E9:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E53:1_1"><span class="lab"><font color="Green" title="E41">A48</font></span></a>, <a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="goboard1.html#T53" title="GOBOARD1:th.53">GOBOARD1:53</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A128</font></span>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c27">mi1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A129</font></span>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c27">mi1</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c27">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E12:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A128</font></span></a>);<br/></span><span class="kw">consider </span><font color="Maroon" title="c28">k2</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> <span class="kw"> such that </span><br/><a NAME="E16:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A130</font></span>: 
<font color="Maroon" title="c28">k2</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="c26">Lma</font>
 <span class="kw">and </span><br/><a NAME="E17:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A131</font></span>: 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">mi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c28">k2</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c29">h2</font> being   <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><span class="kw"> such that </span><br/><a NAME="E19:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A132</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l2</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> holds <br/><font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" title="FINSEQ_4:sch.2">FINSEQ_4:sch 2</a>();<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c30">Ck2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c28">k2</font> as   <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/><a NAME="E21:1_1_5_1_2_1_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A133</font></span>: 
<font color="Maroon" title="c28">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><div><span class="lab"><font color="Green" title="E66">A134</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">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="c29">h2</font> implies <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c5">G1</font> )</span><br/><a NAME="E1:1_1_5_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E67">A135</font></span>: 
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">ma</font>
 <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/><span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E68">A136</font></span>: 
<font color="Maroon" title="c31">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="c29">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c5">G1</font></span><br/><a NAME="E3:1_1_5_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A137</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_4"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E67">A135</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">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_4"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E67">A138</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">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="c29">h2</font> implies  ex <font color="Olive" title="b1">m</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>  ex <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k2</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E68">A139</font></span>: 
<font color="Maroon" title="c31">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="c29">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</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>  ex <font color="Olive" title="b2">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k2</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c32">m</font> = <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">m</font>,<font color="Olive" title="b1">k2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">m</font>,<font color="Olive" title="b1">k2</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c33">k2</font> = <font color="Maroon" title="c28">k2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</font> )</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_5"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E68">A139</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">m</font>,<font color="Maroon" title="c33">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E68">A139</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E68">A140</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c31">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c31">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="c29">h2</font> &amp; <font color="Maroon" title="c31">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="c29">h2</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E69">A141</font></span>: 
<font color="Maroon" title="c31">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="c29">h2</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E70">A142</font></span>: 
<font color="Maroon" title="c31">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="c29">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E70">A142</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A143</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c28">k2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c32">i1</font>, <font color="Maroon" title="c33">i2</font>, <font color="Maroon" title="c34">j1</font>, <font color="Maroon" title="c35">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font> &amp; <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font> implies <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="c32">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c33">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">j2</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="E5:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E72">A144</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E73">A145</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E74">A146</font></span>: 
<font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c32">i1</font>,<font color="Maroon" title="c33">i2</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E75">A147</font></span>: 
<font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c34">j1</font>,<font color="Maroon" title="c35">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c33">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E9:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c31">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E70">A142</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A148</font></span>: 
( <font color="Maroon" title="c34">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c35">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E71">A143</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E73">A145</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E75">A147</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E69">A141</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A149</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span>,<font color="Maroon" title="c28">k2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span>,<font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E69">A141</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_6"/><span class="kw">then </span>
( <font color="Maroon" title="c32">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font> &amp; <font color="Maroon" title="c33">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E77">A149</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E72">A144</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E74">A146</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c32">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c33">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c35">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E76">A148</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E26:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A150</font></span>: 
<font color="Maroon" title="c27">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">mi1</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="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E27:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A151</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E28:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A152</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="ref" href="goboard1.html#D10" title="GOBOARD1:def.10">GOBOARD1:def 10</a></span>;<br/><a NAME="E29:1_1_5_1_2_1_1_2"/>
<font color="Maroon" title="c21">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c27">mi1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c21">ma1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A150</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">f1</font> = <font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><a NAME="E31:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A153</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c33">f</font> being   <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><span class="kw"> such that </span><br/><a NAME="E33:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A154</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">l</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S1" title="FINSEQ_4:sch.1">FINSEQ_4:sch 1</a>(<a class="txt" href="#E31:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A153</font></span></a>);<br/></span><a NAME="E34:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E74">A155</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">mi1</font>
 ;<br/><a NAME="E35:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E75">A156</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">ff</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font>;<br/><a NAME="E36:1_1_5_1_2_1_1_2"/>
<font color="Maroon" title="c21">ma1</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> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E37:1_1_5_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">ma1</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> <font color="Maroon" title="c27">mi1</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="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A157</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><div><span class="lab"><font color="Green" title="E77">A158</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_7"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_7_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_7_1_1"/><span class="lab"><font color="Green" title="E78">A159</font></span>: 
<font color="Maroon" title="c15">mi</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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_7_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">ma1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E67:1_1"><span class="lab"><font color="Green" title="E50">A76</font></span></a>, <a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A160</font></span>: 
<font color="Maroon" title="c21">ma1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_7_1_1"/><span class="lab"><font color="Green" title="E80">A161</font></span>: 
<font color="Maroon" title="c25">w</font> <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="c21">ma1</font>
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_7_1_1"/><span class="lab"><font color="Green" title="E81">A162</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_7_1_1"/>
<font color="Maroon" title="c24">h1</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="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E78">A159</font></span></a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_7_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c33">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="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E81">A162</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E81">A162</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E80">A161</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E79">A160</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E78">A159</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_7_1_2"/><span class="lab"><font color="Green" title="E78">A163</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_7_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E78">A163</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_7_1_2"/><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="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A164</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A165</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> <font color="Maroon" title="c24">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_7_1_2"/><span class="lab"><font color="Green" title="E81">A166</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A167</font></span>: 
<font color="Maroon" title="c22">k1</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="c11">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_7_1_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E81">A166</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A168</font></span>: 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_7_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c33">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E79">A164</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_7_1_2"/><span class="kw">then </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">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</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#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E80">A165</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c22">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E80">A165</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E83">A168</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E15:1_1_5_1_2_1_1_2_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E82">A167</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E40:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A169</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">g1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_8"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_5_1_2_1_1_2_8"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c21">ma1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A170</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">ma1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c35">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> &amp; <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">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">g1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_8"/><span class="lab"><font color="Green" title="E80">A171</font></span>: 
<font color="Maroon" title="c35">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> &amp; <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">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">g1</font> )</span><br/>

<a NAME="E4:1_1_5_1_2_1_1_2_8"/>
<font color="Maroon" title="c35">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E80">A171</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">ma1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">ma1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A172</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">ma1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">mi1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_8"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c35">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E80">A171</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_8"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c21">ma1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c35">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E79">A170</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_8"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c27">mi1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E81">A172</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:1_1_5_1_2_1_1_2_8"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> &amp; <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">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">g1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E79">A173</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">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="c33">f</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="E1:1_1_5_1_2_1_1_2_9"/><span class="lab"><font color="Green" title="E80">A174</font></span>: 
<font color="Maroon" title="c35">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="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#M2" title="SUBSET_1:mode.2">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="E2:1_1_5_1_2_1_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">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">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">i</font>, <font color="Maroon" title="c37">j</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_9"/><span class="lab"><font color="Green" title="E81">A175</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i</font>,<font color="Maroon" title="c37">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i</font>,<font color="Maroon" title="c37">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c38">i</font> = <font color="Maroon" title="c36">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">i</font>,<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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">i</font>,<font color="Olive" title="b1">j</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c39">j</font> = <font color="Maroon" title="c37">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">i</font>,<font color="Maroon" title="c39">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">i</font>,<font color="Maroon" title="c39">j</font> )</span><br/><a NAME="E5:1_1_5_1_2_1_1_2_9"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E80">A174</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_5_1_2_1_1_2_9"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">i</font>,<font color="Maroon" title="c39">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">i</font>,<font color="Maroon" title="c39">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E80">A174</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E81">A175</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E42:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E80">A176</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E43:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E81">A177</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c35">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font> or <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_10"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">n</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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_10"/><span class="lab"><font color="Green" title="E82">A178</font></span>: 
<font color="Maroon" title="c36">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="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_10"/><span class="lab"><font color="Green" title="E83">A179</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_10"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E42:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A176</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E82">A178</font></span></a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A180</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E42:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A176</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E82">A178</font></span></a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_10"/>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">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">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E42:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A176</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E82">A178</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_10"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E83">A179</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E84">A180</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E44:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E82">A181</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E83">A182</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11_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="E1:1_1_5_1_2_1_1_2_11_1"/>
( <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_11_1_1"/><span class="lab"><font color="Green" title="E84">A183</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_11_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A184</font></span>: 
<font color="Maroon" title="c27">mi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_11_1_1"/><span class="lab"><font color="Green" title="E86">A185</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</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="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E42:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A176</font></span></a>, <a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_11_1_1"/>
<font color="Maroon" title="c29">h2</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="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E84">A183</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A186</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E87">A186</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E86">A185</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">mi1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E34:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A155</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E86">A185</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">mi1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E85">A184</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E84">A183</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E84">A187</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_11_1_2"/>
<font color="Maroon" title="c16">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E84">A187</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A188</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A189</font></span>: 
<font color="Maroon" title="c20">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E87">A190</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</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="c29">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E85">A188</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E88">A191</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A192</font></span>: 
<font color="Maroon" title="c28">k2</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="c12">Ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_11_1_2"/>
<font color="Maroon" title="c28">k2</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="c12">Ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E88">A191</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A193</font></span>: 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">l2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E87">A190</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">l2</font></span>)</span>,<font color="Maroon" title="c28">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E44:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E82">A181</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E86">A189</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E90">A193</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E13:1_1_5_1_2_1_1_2_11_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E89">A192</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E46:1_1_5_1_2_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E47:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A194</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="lab"><font color="Green" title="E85">A195</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<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="E49:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E86">A196</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_13"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E87">A197</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c28">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E88">A198</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c22">k1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E89">A199</font></span>: 
<font color="Maroon" title="c25">w</font> <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="c21">ma1</font>
 
;<br/>
<span class="kw">let </span><font color="Maroon" title="c35">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E90">A200</font></span>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E91">A201</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></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><br/>

<span class="kw">set </span><font color="Maroon" title="c36">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font>;<br/>
<span class="kw">assume </span><a NAME="E6:1_1_5_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E92">A202</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c37">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_13"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A195</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E93">A203</font></span>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E91">A201</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_13_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_13_2_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="E1:1_1_5_1_2_1_1_2_13_2_1"/>
( <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E92">A202</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E93">A203</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_13_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E94">A204</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">i</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E95">A205</font></span>: 
<font color="Maroon" title="c38">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">h1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E96">A206</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E97">A207</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">i</font>,<font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E95">A205</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E96">A206</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">y</font> = <font color="Maroon" title="c37">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E96">A206</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E98">A208</font></span>: 
<font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E99">A209</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E94">A204</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E100">A210</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c22">k1</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E88">A198</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E11:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A211</font></span>: 
<font color="Maroon" title="c23">Ck1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Ck1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E102">A212</font></span>: 
<font color="Maroon" title="c38">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">Ck1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E95">A205</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E100">A210</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_13_2_1_1"/>
<font color="Maroon" title="c23">Ck1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Ck1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E95">A205</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E100">A210</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Ck1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E97">A207</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E100">A210</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E102">A212</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c39">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c23">Ck1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E102">A212</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A213</font></span>: 
<font color="Maroon" title="c22">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E90">A200</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E99">A209</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="lab"><font color="Green" title="E104">A214</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c33">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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E89">A199</font></span></a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">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="c23">Ck1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E58:1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E98">A208</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E101">A211</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_13_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A215</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E100">A210</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E103">A213</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_13_2_1_1"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E104">A214</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E22:1_1_5_1_2_1_1_2_13_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E91">A201</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_13_2_1_1"><span class="lab"><font color="Green" title="E105">A215</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_13_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E94">A216</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">i</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E95">A217</font></span>: 
<font color="Maroon" title="c38">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c29">h2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E96">A218</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E97">A219</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i</font></span>)</span>,<font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E95">A217</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E96">A218</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">y</font> = <font color="Maroon" title="c37">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E96">A218</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E98">A220</font></span>: 
<font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E99">A221</font></span>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E94">A216</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E100">A222</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c28">k2</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E87">A197</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E11:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A223</font></span>: 
<font color="Maroon" title="c30">Ck2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Ck2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E102">A224</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">Ck2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E95">A217</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E100">A222</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_13_2_1_2"/>
<font color="Maroon" title="c30">Ck2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Ck2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E95">A217</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E100">A222</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c39">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Ck2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E97">A219</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E100">A222</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E102">A224</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c39">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c30">Ck2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E102">A224</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A225</font></span>: 
<font color="Maroon" title="c28">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E90">A200</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E99">A221</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="lab"><font color="Green" title="E104">A226</font></span>: 
<font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">l</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">Ck2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A131</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E98">A220</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E101">A223</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_13_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A227</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c35">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E100">A222</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E103">A225</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_13_2_1_2"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c31">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E104">A226</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E22:1_1_5_1_2_1_1_2_13_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E91">A201</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_13_2_1_2"><span class="lab"><font color="Green" title="E105">A227</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:1_1_5_1_2_1_1_2_13"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E50:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c35">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E87">A228</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c29">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c36">n1</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E88">A229</font></span>: 
<font color="Maroon" title="c36">n1</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="c29">h2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E89">A230</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E90">A231</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c36">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E88">A229</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_14"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E88">A229</font></span></a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A232</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n1</font></span>)</span>,<font color="Maroon" title="c28">k2</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> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_14"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E87">A228</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">n2</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> <span class="kw"> such that </span><br/><a NAME="E12:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E92">A233</font></span>: 
<font color="Maroon" title="c37">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E93">A234</font></span>: 
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c38">i1</font>, <font color="Maroon" title="c39">i2</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> <span class="kw"> such that </span><br/><a NAME="E15:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E94">A235</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">i1</font>,<font color="Maroon" title="c39">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E16:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E95">A236</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">i1</font>,<font color="Maroon" title="c39">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E92">A233</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c40">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c38">i1</font> as   <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/>
<a NAME="E18:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E96">A237</font></span>: 
<font color="Maroon" title="c39">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E94">A235</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E97">A238</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c40">L</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c40">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c40">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E96">A237</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E95">A236</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E96">A237</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A239</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c40">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E96">A237</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E97">A238</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E92">A233</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E24:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A240</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c40">L</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E98">A239</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E25:1_1_5_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E100">A241</font></span>: 
<font color="Maroon" title="c38">i1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E94">A235</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E26:1_1_5_1_2_1_1_2_14"/>
<font color="Maroon" title="c35">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n1</font></span>)</span>,<font color="Maroon" title="c28">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E88">A229</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E89">A230</font></span></a></span>;<br/>
<a NAME="E27:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c38">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E93">A234</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E94">A235</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E95">A236</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E91">A232</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<a NAME="E28:1_1_5_1_2_1_1_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c36">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E25:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E100">A241</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E99">A240</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:1_1_5_1_2_1_1_2_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E90">A231</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="#E48:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A195</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A111</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>

;<br/><a NAME="E52:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A242</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a>, <a class="txt" href="#E43:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A177</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><a NAME="E53:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E88">A243</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="ref" href="goboard1.html#D10" title="GOBOARD1:def.10">GOBOARD1:def 10</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A244</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><div><span class="lab"><font color="Green" title="E90">A245</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_17"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c35">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="c33">f</font> &amp; <font color="Maroon" title="c35">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="c33">f</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_17"/><span class="lab"><font color="Green" title="E91">A246</font></span>: 
<font color="Maroon" title="c35">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="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_17"/><span class="lab"><font color="Green" title="E92">A247</font></span>: 
<font color="Maroon" title="c35">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="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E91">A246</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A248</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E91">A246</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c35">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E92">A247</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A249</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <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="c6">g1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E92">A247</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c36">i1</font>, <font color="Maroon" title="c37">i2</font>, <font color="Maroon" title="c38">j1</font>, <font color="Maroon" title="c39">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font> implies <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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E7:1_1_5_1_2_1_1_2_17"/><span class="lab"><font color="Green" title="E95">A250</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c35">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c35">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">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E91">A246</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_17"/>
<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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</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="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E93">A248</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E94">A249</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_17"><span class="lab"><font color="Green" title="E95">A250</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c35">hf</font> = <font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font>;<br/><span class="lab"><font color="Green" title="E91">A251</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c29">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/><div><span class="lab"><font color="Green" title="E92">A252</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_19"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c36">i1</font>, <font color="Maroon" title="c37">i2</font>, <font color="Maroon" title="c38">j1</font>, <font color="Maroon" title="c39">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font> &amp; <font color="Maroon" title="c29">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> &amp; 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="c29">h2</font> implies <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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</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="E1:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E93">A253</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E94">A254</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E95">A255</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E96">A256</font></span>: 
<font color="Maroon" title="c29">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_19"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E97">A257</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> <font color="Maroon" title="c29">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:1_1_5_1_2_1_1_2_19"/>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E97">A257</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A258</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c28">k2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E99">A259</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c28">k2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E100">A260</font></span>: 
<font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A130</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_19"/><span class="lab"><font color="Green" title="E101">A261</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</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="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c33">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="c33">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E101">A261</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c31">l</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E101">A261</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">mi1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E101">A261</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c28">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A131</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E100">A260</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E13:1_1_5_1_2_1_1_2_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A262</font></span>: 
( <font color="Maroon" title="c36">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> &amp; <font color="Maroon" title="c37">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E93">A253</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E95">A255</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E99">A259</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_19"/>
<font color="Maroon" title="c29">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c28">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E97">A257</font></span></a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_19"/><span class="kw">then </span>
( <font color="Maroon" title="c38">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c39">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">k2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E94">A254</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E96">A256</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E98">A258</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> = 
<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="c16">ma</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_19"><span class="lab"><font color="Green" title="E102">A262</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_20"><span class="kw">now </span></a><div class="add"><a NAME="E1:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E93">A263</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c22">k1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E94">A264</font></span>: 
<font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c36">i1</font>, <font color="Maroon" title="c37">i2</font>, <font color="Maroon" title="c38">j1</font>, <font color="Maroon" title="c39">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</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="c24">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font> &amp; <font color="Maroon" title="c33">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</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="c24">h1</font> &amp; 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="c33">f</font> implies <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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</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="E3:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E95">A265</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E96">A266</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E97">A267</font></span>: 
<font color="Maroon" title="c24">h1</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="c24">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c36">i1</font>,<font color="Maroon" title="c37">i2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E98">A268</font></span>: 
<font color="Maroon" title="c33">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c38">j1</font>,<font color="Maroon" title="c39">j2</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E99">A269</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">h1</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="c24">h1</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E100">A270</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> <font color="Maroon" title="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E9:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon" title="c19">l1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E99">A269</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A271</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c22">k1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_20"/><span class="lab"><font color="Green" title="E102">A272</font></span>: 
<font color="Maroon" title="c25">w</font> <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="c21">ma1</font>
 ;<br/><a NAME="E12:1_1_5_1_2_1_1_2_20"/><span class="kw">then </span>
<font color="Maroon" title="c33">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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E100">A270</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E100">A270</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E102">A272</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c22">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E94">A264</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E14:1_1_5_1_2_1_1_2_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A273</font></span>: 
( <font color="Maroon" title="c38">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> &amp; <font color="Maroon" title="c39">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E96">A266</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E98">A268</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E93">A263</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon" title="c24">h1</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="c24">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c22">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E99">A269</font></span></a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_20"/><span class="kw">then </span>
( <font color="Maroon" title="c36">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l1</font> &amp; <font color="Maroon" title="c37">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E95">A265</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E97">A267</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E101">A271</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c36">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c37">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c39">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">mi</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_20"><span class="lab"><font color="Green" title="E103">A273</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E59:1_1_5_1_2_1_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><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" title="b2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">j2</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="#E7:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A101</font></span></a>, <a class="txt" href="#E55:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A245</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><a NAME="E60:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A274</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><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" title="b2">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">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="#E25:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A140</font></span></a>, <a class="txt" href="#E57:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A252</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_21"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c36">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c36">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="c24">h1</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">k1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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">k1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">k1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_21"/><span class="lab"><font color="Green" title="E94">A275</font></span>: 
<font color="Maroon" title="c36">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="c24">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">k1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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">k1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">k1</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c37">i</font> = <font color="Maroon" title="c36">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c37">i</font>,<font color="Olive" title="b1">k1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c37">i</font>,<font color="Olive" title="b1">k1</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c38">k1</font> = <font color="Maroon" title="c22">k1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c37">i</font>,<font color="Maroon" title="c38">k1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c37">i</font>,<font color="Maroon" title="c38">k1</font> )</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_21"/>
<font color="Maroon" title="c36">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A96</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_21"><span class="lab"><font color="Green" title="E94">A275</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_21"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c37">i</font>,<font color="Maroon" title="c38">k1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c24">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c37">i</font>,<font color="Maroon" title="c38">k1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1"><span class="lab"><font color="Green" title="E49">A75</font></span></a>, <a class="txt" href="#E60:1_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_21"><span class="lab"><font color="Green" title="E94">A275</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E62:1_1_5_1_2_1_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><a NAME="E63:1_1_5_1_2_1_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A138</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><a NAME="E64:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A276</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E60:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E93">A274</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E65:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E95">A277</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E66:1_1_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A278</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A251</font></span></a>, <a class="txt" href="#E47:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A194</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E67:1_1_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E97">A279</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">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A251</font></span></a>, <a class="txt" href="#E47:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A194</font></span></a>, <a class="txt" href="#E65:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E95">A277</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">thus </span><a NAME="E68:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_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="E1:1_1_5_1_2_1_1_2_22_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E98">A280</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c36">D</font> =  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></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="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E49:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A196</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E98">A280</font></span></a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A281</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A282</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E26:1_1"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E64:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E94">A276</font></span></a>, <a class="txt" href="#E39:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A158</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E67:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E97">A279</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">GOBOARD1:41</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E101">A283</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E27:1_1"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="txt" href="#E45:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A182</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E53:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A243</font></span></a>, <a class="txt" href="#E66:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E96">A278</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E99">A281</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>7</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> );<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E102">A284</font></span>: 
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[1]</span><br/>

<span class="kw">thus </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1"><span class="lab"><font color="Green" title="E11">A10</font></span></a>, <a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E33:1_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T22" title="GOBOARD1:th.22">GOBOARD1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E103">A285</font></span>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c37">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E104">A286</font></span>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c37">m</font>] &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">m</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" title="NAT_1:sch.6">NAT_1:sch 6</a>(<a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E103">A285</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E102">A284</font></span></a>);<br/></span><a NAME="E10:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E105">A287</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c37">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E104">A286</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">m</font> = <font color="Maroon" title="c37">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">p</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">m</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><div><span class="lab"><font color="Green" title="E106">A288</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_2"/><span class="lab"><font color="Green" title="E107">A289</font></span>: 
<font color="Maroon" title="c38">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_2"/>
<font color="Maroon" title="c38">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E104">A286</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c39">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_2"><span class="lab"><font color="Green" title="E107">A289</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_2"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <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="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E104">A286</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2"><span class="lab"><font color="Green" title="E51">A92</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">ll</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">m</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><span class="kw">consider </span><font color="Maroon" title="c41">t</font> being   <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><span class="kw"> such that </span><br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E107">A290</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">ll</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">t</font> holds <br/><font color="Maroon" title="c41">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" title="FINSEQ_4:sch.2">FINSEQ_4:sch 2</a>();<br/></span><a NAME="E17:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E108">A291</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c40">ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E109">A292</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c42">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">t</font> or <font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c43">x</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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="lab"><font color="Green" title="E110">A293</font></span>: 
<font color="Maroon" title="c43">x</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="c41">t</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="lab"><font color="Green" title="E111">A294</font></span>: 
<font color="Maroon" title="c41">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon" title="c43">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E108">A291</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E110">A293</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A295</font></span>: 
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c40">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="lab"><font color="Green" title="E113">A296</font></span>: 
<font color="Maroon" title="c43">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">m</font>
 
<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="E8:1_1_5_1_2_1_1_2_22_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E108">A291</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E110">A293</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E113">A296</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E112">A295</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon" title="c42">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E110">A293</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_3"><span class="lab"><font color="Green" title="E111">A294</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E110">A297</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">t</font> holds <br/><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c42">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c42">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="c41">t</font> implies <font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )</span><br/>

<a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_4"/><span class="lab"><font color="Green" title="E111">A298</font></span>: 
<font color="Maroon" title="c42">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">m</font>
 
<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/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_4"/><span class="lab"><font color="Green" title="E112">A299</font></span>: 
<font color="Maroon" title="c42">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="c41">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c42">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c40">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E108">A291</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A300</font></span>: 
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c40">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E108">A291</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_1_4"><span class="lab"><font color="Green" title="E112">A299</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_4"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c42">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_4"><span class="lab"><font color="Green" title="E111">A298</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">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_4"/>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c42">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_4"><span class="lab"><font color="Green" title="E113">A300</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E111">A301</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c42">t</font> = <font color="Maroon" title="c41">t</font> as   <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/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E112">A302</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2"><span class="lab"><font color="Green" title="E51">A92</font></span></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">GOBOARD1:26</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E113">A303</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span></span>)</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="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E114">A304</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">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="c42">t</font> implies  ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">l</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">l</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E115">A305</font></span>: 
<font color="Maroon" title="c43">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="c42">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">l</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">l</font> )</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E110">A297</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c44">i1</font>, <font color="Maroon" title="c45">i2</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E116">A306</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E117">A307</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E118">A308</font></span>: 
<font color="Maroon" title="c45">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E116">A306</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A309</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c46">l</font> = <font color="Maroon" title="c45">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c47">Ci2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c45">i2</font> as   <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/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E120">A310</font></span>: 
<font color="Maroon" title="c44">i1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E116">A306</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_1_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c47">Ci2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A311</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c47">Ci2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c47">Ci2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c44">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">Ci2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c44">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E120">A310</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c47">Ci2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c44">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E120">A310</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A312</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c45">i2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E117">A307</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E120">A310</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E121">A311</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_5_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_5_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E108">A291</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E115">A305</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_5_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A313</font></span>: 
<font color="Maroon" title="c38">m</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> <font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_5_2"/>
<font color="Maroon" title="c45">i2</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="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E104">A286</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E110">A297</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E115">A305</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E122">A312</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_5_2"/><span class="kw">then </span>
<font color="Maroon" title="c38">m</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> <font color="Maroon" title="c38">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_1_5_2"><span class="lab"><font color="Green" title="E123">A313</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_5_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c38">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c38">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_5_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_5_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E119">A309</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1_5"/><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="c45">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A314</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c48">i1</font> = <font color="Maroon" title="c44">i1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i1</font>,<font color="Olive" title="b1">l</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Olive" title="b1">l</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c49">l</font> = <font color="Maroon" title="c46">l</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">l</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">l</font> )</span><br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="lab"><font color="Green" title="E124">A315</font></span>: 
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E115">A305</font></span></a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_1_5"/>
<font color="Maroon" title="c45">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E118">A308</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A316</font></span>: 
<font color="Maroon" title="c49">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c49">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E123">A314</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">hence </span><a NAME="E24:1_1_5_1_2_1_1_2_22_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">l</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E113">A303</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E120">A310</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">l</font></span><br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_1_5"/>
<font color="Maroon" title="c49">l</font> <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="c45">i2</font>
 ;<br/><span class="kw">hence </span><a NAME="E26:1_1_5_1_2_1_1_2_22_1_1_5"/>
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c48">i1</font>,<font color="Maroon" title="c49">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E117">A307</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E120">A310</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E124">A315</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E123">A314</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1_5"><span class="lab"><font color="Green" title="E125">A316</font></span></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">GOBOARD1:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E25:1_1_5_1_2_1_1_2_22_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A317</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A318</font></span>: 
(  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="ref" href="goboard1.html#T30" title="GOBOARD1:th.30">GOBOARD1:30</a></span>;<br/><a NAME="E29:1_1_5_1_2_1_1_2_22_1_1"/>
<font color="Maroon" title="c38">m</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E106">A288</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A319</font></span>: 
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="c42">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c40">ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E32:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">t</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"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E111">A301</font></span></a></span>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A320</font></span>: 
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E104">A286</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E105">A287</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E116">A318</font></span></a>, <a class="ref" href="goboard1.html#T48" title="GOBOARD1:th.48">GOBOARD1:48</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c43">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c43">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="c42">t</font> &amp; <font color="Maroon" title="c43">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="c42">t</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/>1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E119">A321</font></span>: 
<font color="Maroon" title="c43">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="c42">t</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E120">A322</font></span>: 
<font color="Maroon" title="c43">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="c42">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/>1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c44">i1</font>, <font color="Maroon" title="c45">i2</font>, <font color="Maroon" title="c46">j1</font>, <font color="Maroon" title="c47">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">j1</font>,<font color="Maroon" title="c47">j2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</font> &amp; <font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c46">j1</font>,<font color="Maroon" title="c47">j2</font> implies 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c44">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c45">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c47">j2</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E121">A323</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E122">A324</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">j1</font>,<font color="Maroon" title="c47">j2</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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E123">A325</font></span>: 
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">i1</font>,<font color="Maroon" title="c45">i2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E124">A326</font></span>: 
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c46">j1</font>,<font color="Maroon" title="c47">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c44">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c45">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c47">j2</font></span>)</span></span>)</span></span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E125">A327</font></span>: 
<font color="Maroon" title="c44">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E113">A303</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E121">A323</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c45">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E113">A303</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E121">A323</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A328</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">i2</font> &amp; <font color="Maroon" title="c45">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c45">i2</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E125">A327</font></span></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">GOBOARD1:32</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A329</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c44">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c45">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E125">A327</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E119">A321</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A330</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c44">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c45">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E123">A325</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E125">A327</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E126">A328</font></span></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">GOBOARD1:32</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A331</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E110">A297</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E120">A322</font></span></a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="lab"><font color="Green" title="E130">A332</font></span>: 
<font color="Maroon" title="c46">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E113">A303</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E122">A324</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c47">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E113">A303</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E122">A324</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A333</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c47">j2</font> &amp; <font color="Maroon" title="c47">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c47">j2</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E125">A327</font></span></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">GOBOARD1:32</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A334</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c46">j1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c47">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E130">A332</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c42">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c43">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"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c43">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E120">A322</font></span></a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A335</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font></span>)</span> <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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c46">j1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c47">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E54:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A244</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E124">A326</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E130">A332</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E131">A333</font></span></a>, <a class="ref" href="goboard1.html#T32" title="GOBOARD1:th.32">GOBOARD1:32</a></span>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c43">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E110">A297</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E119">A321</font></span></a></span>;<br/><span class="kw">hence </span>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="c44">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c45">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c47">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E128">A330</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E133">A335</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E127">A329</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E132">A334</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_1_6"><span class="lab"><font color="Green" title="E129">A331</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>
<br/>.= 
<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="c44">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c45">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c47">j2</font></span>)</span></span>)</span>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A336</font></span>: 
<font color="Maroon" title="c42">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E114">A304</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c43">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">t</font></span>)</span>;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_1"/><span class="lab"><font color="Green" title="E120">A337</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E109">A292</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c40">ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E30:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E117">A319</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c42">t</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="c42">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c40">ll</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E107">A290</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E111">A301</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>

;<br/><a NAME="E40:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c42">t</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="c42">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E32:1_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E115">A317</font></span></a>, <a class="ref" href="goboard1.html#T30" title="GOBOARD1:th.30">GOBOARD1:30</a></span>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">t</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A251</font></span></a>, <a class="txt" href="#E47:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A194</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E112">A302</font></span></a>, <a class="txt" href="#E30:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E117">A319</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E118">A320</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E119">A336</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E100">A282</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E101">A283</font></span></a></span>;<br/><a NAME="E42:1_1_5_1_2_1_1_2_22_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c43">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1_5_1_2_1_1_2_22_1_1"><span class="lab"><font color="Green" title="E120">A337</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E43:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A242</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2"/><span class="lab"><font color="Green" title="E98">A338</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c36">D</font> =  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2"/><span class="lab"><font color="Green" title="E99">A339</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c4">k</font> <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="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></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="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E49:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A196</font></span></a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A340</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A341</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E64:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E94">A276</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">GOBOARD1:41</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c37">t</font> being   <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><span class="kw"> such that </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="lab"><font color="Green" title="E102">A342</font></span>: 
( <font color="Maroon" title="c37">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon" title="c37">t</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="c37">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 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="c37">t</font> &amp; <font color="Maroon" title="c37">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A343</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E33:1_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E16:1_1"><span class="lab"><font color="Green" title="E15">A14</font></span></a>, <a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2"><span class="lab"><font color="Green" title="E99">A339</font></span></a>, <a class="ref" href="goboard1.html#T51" title="GOBOARD1:th.51">GOBOARD1:51</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c38">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font></span>)</span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A344</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A343</font></span></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A345</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2"><span class="lab"><font color="Green" title="E51">A92</font></span></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">GOBOARD1:26</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E26:1_1"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E27:1_1"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="txt" href="#E39:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A158</font></span></a>, <a class="txt" href="#E45:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A182</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A152</font></span></a>, <a class="txt" href="#E67:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E97">A279</font></span></a>, <a class="txt" href="#E66:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E96">A278</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A340</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A251</font></span></a>, <a class="txt" href="#E47:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A194</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A342</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A345</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A341</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c38">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c24">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c29">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A344</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A242</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E100">A346</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E101">A347</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="txt" href="#E55:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A245</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">t</font> being   <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><span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E102">A348</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E103">A349</font></span>: 
( <font color="Maroon" title="c37">t</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c37">t</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="c37">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; 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="c37">t</font> &amp; <font color="Maroon" title="c37">t</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2"><span class="lab"><font color="Green" title="E98">A338</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A346</font></span></a>, <a class="ref" href="goboard1.html#T52" title="GOBOARD1:th.52">GOBOARD1:52</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c38">tt</font> being   <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><span class="kw"> such that </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A350</font></span>: 
( <font color="Maroon" title="c38">tt</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon" title="c38">tt</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="c38">tt</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 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="c38">tt</font> &amp; <font color="Maroon" title="c38">tt</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E105">A351</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2"><span class="lab"><font color="Green" title="E99">A339</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A349</font></span></a>, <a class="ref" href="goboard1.html#T51" title="GOBOARD1:th.51">GOBOARD1:51</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E106">A352</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c26">Lma</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c39">lg</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>7</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span> );<br/><span class="kw">defpred </span><font color="Maroon">S<sub>8</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </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">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">ma</font></span>)</span> );<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E107">A353</font></span>: 
<font color="Maroon" title="c39">lg</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></span>;<br/><div><span class="lab"><font color="Green" title="E108">A354</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c40">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
<font color="Maroon" title="c40">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c41">n</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E109">A355</font></span>: 
( <font color="Maroon" title="c41">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c40">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c41">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c42">n</font> = <font color="Maroon" title="c41">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c43">n</font> = <font color="Maroon" title="c42">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c43">n</font>]</span><br/><span class="kw">thus </span><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c43">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E109">A355</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c40">pf</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E109">A356</font></span>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c40">pf</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c40">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E108">A354</font></span></a>);<br/></span><span class="kw">defpred </span><font color="Maroon">S<sub>9</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( $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="c33">f</font> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> );<br/><span class="kw">reconsider </span><font color="Maroon" title="c41">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>, <font color="Maroon" title="c42">Li</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font>, <font color="Maroon" title="c43">La</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">ma</font> as   <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/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E110">A357</font></span>: 
<font color="Maroon" title="c39">lg</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 ;<br/><div><span class="lab"><font color="Green" title="E111">A358</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c44">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c16">ma</font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
<font color="Maroon" title="c44">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c45">n</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/><span class="lab"><font color="Green" title="E112">A359</font></span>: 
( <font color="Maroon" title="c45">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c44">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c46">n</font> = <font color="Maroon" title="c45">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c47">n</font> = <font color="Maroon" title="c46">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c47">n</font>]</span><br/><span class="kw">thus </span><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c47">n</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E112">A359</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c44">pl</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E112">A360</font></span>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c44">pl</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c44">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E111">A358</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c45">pf</font> = <font color="Maroon" title="c40">pf</font>, <font color="Maroon" title="c46">pl</font> = <font color="Maroon" title="c44">pl</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E113">A361</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c47">K2</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> <span class="kw"> such that </span><br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E114">A362</font></span>: 
<font color="Maroon" title="c47">K2</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="c26">Lma</font>
 <span class="kw">and </span><br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E115">A363</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c48">CK2</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c47">K2</font> as   <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">consider </span><font color="Maroon" title="c49">K1</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> <span class="kw"> such that </span><br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E116">A364</font></span>: 
<font color="Maroon" title="c49">K1</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="c42">Li</font>
 <span class="kw">and </span><br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E117">A365</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Li</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c50">CK1</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">K1</font> as   <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">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> $1,<font color="Maroon" title="c49">K1</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c51">h1</font> being   <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><span class="kw"> such that </span><br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E118">A366</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l1</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">h1</font> holds <br/><font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" title="FINSEQ_4:sch.2">FINSEQ_4:sch 2</a>();<br/></span><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E119">A367</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c52">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c52">k</font>] implies <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E120">A368</font></span>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c52">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>9</sub></font>[<font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E121">A369</font></span>: 
<font color="Maroon" title="c52">k</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="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c53">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">m</font></span>)</span> implies <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"/><span class="lab"><font color="Green" title="E122">A370</font></span>: 
( <font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c53">m</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/>
( <font color="Maroon" title="c52">k</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="c52">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E123">A371</font></span>: 
<font color="Maroon" title="c52">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"/>
( <font color="Maroon" title="c25">w</font> <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="c21">ma1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c33">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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">ma1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">Li</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="txt" href="#E58:1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E123">A371</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E122">A370</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c52">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c52">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A372</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c52">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c52">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E121">A369</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A373</font></span>: 
<font color="Maroon" title="c52">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A374</font></span>: 
<font color="Maroon" title="c52">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="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E123">A372</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c54">i1</font>, <font color="Maroon" title="c55">i2</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> <span class="kw"> such that </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E126">A375</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E127">A376</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E128">A377</font></span>: 
<font color="Maroon" title="c55">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E126">A375</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c56">j1</font>, <font color="Maroon" title="c57">j2</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> <span class="kw"> such that </span><br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E129">A378</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E130">A379</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E121">A369</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c58">Lj1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c56">j1</font> as   <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/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E131">A380</font></span>: 
<font color="Maroon" title="c57">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E129">A378</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E132">A381</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Lj1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c58">Lj1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Lj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c58">Lj1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">Lj1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c57">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E131">A380</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</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> <font color="Maroon" title="c58">Lj1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E130">A379</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E131">A380</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A382</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c58">Lj1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E131">A380</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E132">A381</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E134">A383</font></span>: 
<font color="Maroon" title="c56">j1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E129">A378</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c59">Li1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c54">i1</font> as   <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/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E135">A384</font></span>: 
<font color="Maroon" title="c54">i1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E126">A375</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E136">A385</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c59">Li1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c59">Li1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c59">Li1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A386</font></span>: 
<font color="Maroon" title="c59">Li1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">Li1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E128">A377</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">Li1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E127">A376</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E128">A377</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A387</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c59">Li1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E128">A377</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E136">A385</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A388</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c54">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E120">A368</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E123">A372</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E124">A373</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E135">A384</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">i1</font> or <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c54">i1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E139">A388</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E140">A389</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">i1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E125">A374</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E125">A374</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c59">Li1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E127">A376</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E128">A377</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E137">A386</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c15">mi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E128">A377</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E136">A385</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E140">A389</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A390</font></span>: 
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">ma1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E125">A374</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">w</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> <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E123">A372</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c52">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">ma1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E141">A390</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A391</font></span>: 
<font color="Maroon" title="c21">ma1</font> <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="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">mi</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> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A127</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A392</font></span>: 
<font color="Maroon" title="c15">mi</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:1_1"><span class="lab"><font color="Green" title="E41">A48</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font> <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="#E42:1_1"><span class="lab"><font color="Green" title="E35">A42</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A393</font></span>: 
<font color="Maroon" title="c15">mi</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E143">A392</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/>
( <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E121">A369</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">mi</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="txt" href="#E6:1_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E9:1_1"><span class="lab"><font color="Green" title="E10">A9</font></span></a>, <a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E54:1_1"><span class="lab"><font color="Green" title="E42">A49</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E142">A391</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E144">A393</font></span></a>, <a class="ref" href="goboard1.html#T44" title="GOBOARD1:th.44">GOBOARD1:44</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c53">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> <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="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E122">A370</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E144">A393</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>;<br/><span class="kw">hence </span><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E140">A394</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c54">i1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c54">i1</font></span>)</span> or for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</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="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A347</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E121">A369</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E125">A374</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E135">A384</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E138">A387</font></span></a>, <a class="ref" href="goboard1.html#T43" title="GOBOARD1:th.43">GOBOARD1:43</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c54">i1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E122">A370</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E135">A384</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A394</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</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="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E141">A395</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j1</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="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E134">A383</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E133">A382</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c56">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c54">i1</font> &amp; <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c54">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c56">j1</font> &amp; <font color="Maroon" title="c56">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E141">A395</font></span></a>, <a class="ref" href="seqm_3.html#T81" title="SEQM_3:th.81">SEQM_3:81</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E142">A396</font></span>: 
( <font color="Maroon" title="c56">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c54">i1</font> &amp; <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A394</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E122">A370</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E134">A383</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E133">A382</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E142">A396</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c54">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c56">j1</font> &amp; <font color="Maroon" title="c56">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c56">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A394</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">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E122">A370</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E134">A383</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E133">A382</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1_2"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E120">A397</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Li</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c42">Li</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Li</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><div><span class="lab"><font color="Green" title="E121">A398</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c52">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="c51">h1</font> implies <font color="Maroon" title="c52">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="c5">G1</font> )</span><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E122">A399</font></span>: 
<font color="Maroon" title="c19">l1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T45" title="XREAL_1:th.45">XREAL_1:45</a></span>;<br/><span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E123">A400</font></span>: 
<font color="Maroon" title="c52">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="c51">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c52">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="c5">G1</font></span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A401</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c52">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/>
<font color="Maroon" title="c52">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E123">A400</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A402</font></span>: 
<font color="Maroon" title="c52">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E122">A399</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/>
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c52">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E125">A402</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">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"/>
<font color="Maroon" title="c52">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E124">A401</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E122">A403</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c52">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="c51">h1</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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">K1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">K1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E123">A404</font></span>: 
<font color="Maroon" title="c52">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="c51">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">K1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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">K1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">K1</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c53">i</font> = <font color="Maroon" title="c52">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">K1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i</font>,<font color="Olive" title="b1">K1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i</font>,<font color="Olive" title="b1">K1</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c54">K1</font> = <font color="Maroon" title="c49">K1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">K1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">K1</font> )</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
<font color="Maroon" title="c52">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E123">A404</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">K1</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i</font>,<font color="Maroon" title="c54">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E123">A404</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E123">A405</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c52">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c52">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="c51">h1</font> &amp; <font color="Maroon" title="c52">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="c51">h1</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E124">A406</font></span>: 
<font color="Maroon" title="c52">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="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E125">A407</font></span>: 
<font color="Maroon" title="c52">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="c51">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon" title="c52">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E125">A407</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A408</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c53">i1</font>, <font color="Maroon" title="c54">i2</font>, <font color="Maroon" title="c55">j1</font>, <font color="Maroon" title="c56">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font> &amp; <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font> implies <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="c53">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c54">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j2</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="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E127">A409</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E128">A410</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E129">A411</font></span>: 
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i1</font>,<font color="Maroon" title="c54">i2</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="lab"><font color="Green" title="E130">A412</font></span>: 
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c54">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E125">A407</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A413</font></span>: 
( <font color="Maroon" title="c55">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c56">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E126">A408</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E128">A410</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E130">A412</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon" title="c52">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E124">A406</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A414</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c52">n</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c52">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c52">n</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E124">A406</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><span class="kw">then </span>
( <font color="Maroon" title="c53">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">n</font> &amp; <font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E132">A414</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E127">A409</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E129">A411</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c53">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c54">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c52">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c52">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><span class="lab"><font color="Green" title="E131">A413</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E124">A415</font></span>: 
<font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E125">A416</font></span>: 
<font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A417</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A363</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c47">K2</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c52">h2</font> being   <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><span class="kw"> such that </span><br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E127">A418</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">l2</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c52">h2</font> holds <br/><font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" title="FINSEQ_4:sch.2">FINSEQ_4:sch 2</a>();<br/></span><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E128">A419</font></span>: 
<font color="Maroon">S<sub>9</sub></font>[ <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#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E129">A420</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>9</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E128">A419</font></span></a>, <a class="txt" href="#E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A367</font></span></a>);<br/></span><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E130">A421</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c53">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E131">A422</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c54">i2</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E132">A423</font></span>: 
<font color="Maroon" title="c54">i2</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="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E133">A424</font></span>: 
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">h1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A425</font></span>: 
( <font color="Maroon" title="c19">l1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mi</font> &amp; <font color="Maroon" title="c54">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E132">A423</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon" title="c54">i2</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E132">A423</font></span></a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A426</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i2</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E131">A422</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A351</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c55">i1</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> <span class="kw"> such that </span><br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E136">A427</font></span>: 
<font color="Maroon" title="c55">i1</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="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E137">A428</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A348</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c56">n1</font>, <font color="Maroon" title="c57">n2</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> <span class="kw"> such that </span><br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E138">A429</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">n1</font>,<font color="Maroon" title="c57">n2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E139">A430</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">n1</font>,<font color="Maroon" title="c57">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E136">A427</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c58">Ln1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c56">n1</font> as   <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/>
<a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E140">A431</font></span>: 
<font color="Maroon" title="c57">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E138">A429</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="lab"><font color="Green" title="E141">A432</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Ln1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c58">Ln1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Ln1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c58">Ln1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">Ln1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c57">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E140">A431</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">Ln1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E139">A430</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E140">A431</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A433</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c58">Ln1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E140">A431</font></span></a>, <a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E141">A432</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon" title="c56">n1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E138">A429</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A434</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c56">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E129">A420</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E136">A427</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E142">A433</font></span></a></span>;<br/>
<a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i2</font>,<font color="Maroon" title="c49">K1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E132">A423</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E133">A424</font></span></a></span>;<br/>
<a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E137">A428</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E138">A429</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E139">A430</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E135">A426</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E143">A434</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><span class="lab"><font color="Green" title="E134">A425</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">defpred </span><font color="Maroon">S<sub>10</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( $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="c33">f</font> implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font> );<br/><a NAME="E45:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E131">A435</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c53">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>10</sub></font>[<font color="Maroon" title="c53">k</font>] implies <font color="Maroon">S<sub>10</sub></font>[<font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><span class="lab"><font color="Green" title="E132">A436</font></span>: 
<font color="Maroon">S<sub>10</sub></font>[<font color="Maroon" title="c53">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>10</sub></font>[<font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><span class="lab"><font color="Green" title="E133">A437</font></span>: 
<font color="Maroon" title="c53">k</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="c33">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">m</font></span>)</span> holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c54">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c54">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c54">m</font></span>)</span> implies <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><span class="lab"><font color="Green" title="E134">A438</font></span>: 
( <font color="Maroon" title="c54">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c54">m</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1"/>
( <font color="Maroon" title="c53">k</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="c53">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"/><span class="lab"><font color="Green" title="E135">A439</font></span>: 
<font color="Maroon" title="c53">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c31">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A157</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c33">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"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</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> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">Li</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="txt" href="#E58:1_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"><span class="lab"><font color="Green" title="E135">A439</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A126</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E134">A438</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E135">A440</font></span>: 
<font color="Maroon" title="c53">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E136">A441</font></span>: 
<font color="Maroon" title="c53">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A176</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E133">A437</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A442</font></span>: 
<font color="Maroon" title="c53">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c55">j1</font>, <font color="Maroon" title="c56">j2</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E138">A443</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E139">A444</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c55">j1</font>,<font color="Maroon" title="c56">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E133">A437</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c57">Lj1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c55">j1</font> as   <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/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E140">A445</font></span>: 
<font color="Maroon" title="c56">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E138">A443</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E141">A446</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c57">Lj1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c57">Lj1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c57">Lj1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c57">Lj1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c57">Lj1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c56">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E140">A445</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</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> <font color="Maroon" title="c57">Lj1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E139">A444</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E140">A445</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A447</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c57">Lj1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E140">A445</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E141">A446</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E143">A448</font></span>: 
<font color="Maroon" title="c55">j1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E138">A443</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A449</font></span>: 
<font color="Maroon" title="c55">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c54">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E134">A438</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E142">A447</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E135">A440</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E145">A450</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A451</font></span>: 
<font color="Maroon" title="c53">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="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E137">A442</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c58">i1</font>, <font color="Maroon" title="c59">i2</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> <span class="kw"> such that </span><br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E147">A452</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c58">i1</font>,<font color="Maroon" title="c59">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E148">A453</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c58">i1</font>,<font color="Maroon" title="c59">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c60">Li1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c58">i1</font> as   <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/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E149">A454</font></span>: 
<font color="Maroon" title="c59">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E147">A452</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E150">A455</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c60">Li1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c60">Li1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c60">Li1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c60">Li1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c60">Li1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c59">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E149">A454</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c60">Li1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E148">A453</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E149">A454</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E151">A456</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c60">Li1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E149">A454</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E150">A455</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="lab"><font color="Green" title="E152">A457</font></span>: 
<font color="Maroon" title="c58">i1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E147">A452</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E153">A458</font></span>: 
<font color="Maroon" title="c58">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E132">A436</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E145">A450</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E137">A442</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E151">A456</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1"/>
( <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">i1</font> or <font color="Maroon" title="c58">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">ma</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E153">A458</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E154">A459</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">i1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E155">A460</font></span>: 
<font color="Maroon" title="c25">w</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> <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E145">A450</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E156">A461</font></span>: 
( <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">g1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c27">mi1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E146">A451</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">ma1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E49:1_1"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A94</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E151">A456</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E154">A459</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">ma1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E155">A460</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A462</font></span>: 
<font color="Maroon" title="c27">mi1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A129</font></span></a>, <a class="txt" href="#E40:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A169</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A156</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E146">A451</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E151">A456</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E154">A459</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E156">A461</font></span></a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E34:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A155</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E137">A442</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c25">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">mi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E157">A462</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A154</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E136">A441</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E154">A463</font></span>: 
<font color="Maroon" title="c58">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c58">i1</font></span>)</span> or for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</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="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c58">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A347</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E133">A437</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E146">A451</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E152">A457</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E151">A456</font></span></a>, <a class="ref" href="goboard1.html#T43" title="GOBOARD1:th.43">GOBOARD1:43</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c58">i1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><span class="lab"><font color="Green" title="E134">A438</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E152">A457</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E154">A463</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive" title="b1">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">l</font></span>)</span> &amp; <font color="Olive" title="b1">l</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="c5">G1</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c58">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A464</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c58">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">j1</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="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E143">A448</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E142">A447</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c55">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c58">i1</font> &amp; <font color="Maroon" title="c58">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c58">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c55">j1</font> &amp; <font color="Maroon" title="c55">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E155">A464</font></span></a>, <a class="ref" href="seqm_3.html#T81" title="SEQM_3:th.81">SEQM_3:81</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c55">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c58">i1</font> &amp; <font color="Maroon" title="c58">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E144">A449</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E154">A463</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c58">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c55">j1</font> &amp; <font color="Maroon" title="c55">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"><span class="lab"><font color="Green" title="E144">A449</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E154">A463</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2_1_1_2"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/>
<font color="Maroon" title="c54">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E132">A465</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E133">A466</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c53">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c53">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="c52">h2</font> implies <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">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="c5">G1</font> )</span><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><span class="lab"><font color="Green" title="E134">A467</font></span>: 
<font color="Maroon" title="c53">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">ma</font>
 <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/><span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><span class="lab"><font color="Green" title="E135">A468</font></span>: 
<font color="Maroon" title="c53">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="c52">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">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="c5">G1</font></span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c53">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A469</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E135">A468</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E134">A467</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">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><span class="lab"><font color="Green" title="E136">A469</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E134">A470</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c53">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c53">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="c52">h2</font> implies  ex <font color="Olive" title="b1">m</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>  ex <font color="Olive" title="b2">K2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">K2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">K2</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/><span class="lab"><font color="Green" title="E135">A471</font></span>: 
<font color="Maroon" title="c53">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="c52">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</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>  ex <font color="Olive" title="b2">K2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">K2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">K2</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c54">m</font> = <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">K2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">m</font>,<font color="Olive" title="b1">K2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">m</font>,<font color="Olive" title="b1">K2</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c55">K2</font> = <font color="Maroon" title="c47">K2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">m</font>,<font color="Maroon" title="c55">K2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">m</font>,<font color="Maroon" title="c55">K2</font> )</span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E135">A471</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">m</font>,<font color="Maroon" title="c55">K2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">m</font>,<font color="Maroon" title="c55">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><span class="lab"><font color="Green" title="E135">A471</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E135">A472</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c53">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c53">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="c52">h2</font> &amp; <font color="Maroon" title="c53">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="c52">h2</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E136">A473</font></span>: 
<font color="Maroon" title="c53">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="c52">h2</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E137">A474</font></span>: 
<font color="Maroon" title="c53">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="c52">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E137">A474</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A475</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c54">i1</font>, <font color="Maroon" title="c55">i2</font>, <font color="Maroon" title="c56">j1</font>, <font color="Maroon" title="c57">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</font> &amp; <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</font> implies <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="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c55">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c57">j2</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="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E139">A476</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E140">A477</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E141">A478</font></span>: 
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">i1</font>,<font color="Maroon" title="c55">i2</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="lab"><font color="Green" title="E142">A479</font></span>: 
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">j1</font>,<font color="Maroon" title="c57">j2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c55">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c57">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E137">A474</font></span></a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A480</font></span>: 
( <font color="Maroon" title="c56">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c57">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E138">A475</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E140">A477</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E142">A479</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E136">A473</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A481</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E136">A473</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><span class="kw">then </span>
( <font color="Maroon" title="c54">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font> &amp; <font color="Maroon" title="c55">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E144">A481</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E139">A476</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E141">A478</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c54">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c56">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c55">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c57">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">n</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><span class="lab"><font color="Green" title="E143">A480</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E50:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E136">A482</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E51:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E137">A483</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E138">A484</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E53:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E139">A485</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c45">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A94</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="goboard1.html#T19" title="GOBOARD1:th.19">GOBOARD1:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1"/>
( <font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">pl</font> or <font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c46">pl</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E139">A485</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_1"/>
<font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">pl</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c45">pf</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E137">A483</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_1"/><span class="kw">then </span>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A361</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">hence </span><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_1"/>
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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_2"/>
<font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c46">pl</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c46">pl</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E124">A415</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_2"/><span class="kw">then </span>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E138">A484</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">hence </span><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12_1_2"/>
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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E55:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E140">A486</font></span>: 
<font color="Maroon" title="c39">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E107">A353</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E141">A487</font></span>: 
<font color="Maroon">S<sub>10</sub></font>[ <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#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E142">A488</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>10</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E141">A487</font></span></a>, <a class="txt" href="#E45:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E131">A435</font></span></a>);<br/></span><a NAME="E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E143">A489</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c53">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E144">A490</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c54">i2</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E145">A491</font></span>: 
<font color="Maroon" title="c54">i2</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="c52">h2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E146">A492</font></span>: 
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c54">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E145">A491</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A493</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c54">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c54">i2</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E145">A491</font></span></a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E148">A494</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c54">i2</font></span>)</span>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E144">A490</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c37">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A351</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c55">i1</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> <span class="kw"> such that </span><br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E149">A495</font></span>: 
<font color="Maroon" title="c55">i1</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="c33">f</font>
 <span class="kw">and </span><br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E150">A496</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A348</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c56">n1</font>, <font color="Maroon" title="c57">n2</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> <span class="kw"> such that </span><br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E151">A497</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c56">n1</font>,<font color="Maroon" title="c57">n2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E152">A498</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c56">n1</font>,<font color="Maroon" title="c57">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A173</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E149">A495</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c58">Ln1</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c56">n1</font> as   <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/>
<a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E153">A499</font></span>: 
<font color="Maroon" title="c57">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E151">A497</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E154">A500</font></span>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Ln1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c58">Ln1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c58">Ln1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c58">Ln1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">Ln1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c57">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E153">A499</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c58">Ln1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c57">n2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E152">A498</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E153">A499</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E155">A501</font></span>: 
<font color="Maroon" title="c33">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c58">Ln1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E153">A499</font></span></a>, <a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E154">A500</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="lab"><font color="Green" title="E156">A502</font></span>: 
<font color="Maroon" title="c56">n1</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="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E151">A497</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<font color="Maroon" title="c53">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c54">i2</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E145">A491</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E146">A492</font></span></a></span>;<br/>
<a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c54">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">n1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E150">A496</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E151">A497</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E152">A498</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E148">A494</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E142">A488</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E149">A495</font></span></a>, <a class="txt" href="#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E156">A502</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E155">A501</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><span class="lab"><font color="Green" title="E147">A493</font></span></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E60:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></span>;<br/><a NAME="E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E144">A503</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E62:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E145">A504</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</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="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1"><span class="lab"><font color="Green" title="E20">A23</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E18:1_1"><span class="lab"><font color="Green" title="E17">A16</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c53">i2</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> <span class="kw"> such that </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E146">A505</font></span>: 
<font color="Maroon" title="c53">i2</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="c41">C</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E147">A506</font></span>: 
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="lab"><font color="Green" title="E148">A507</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A508</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E146">A505</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E146">A505</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A509</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E146">A505</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E147">A506</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E148">A507</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><span class="kw">assume </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E151">A510</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c54">n1</font>, <font color="Maroon" title="c55">n2</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> <span class="kw"> such that </span><br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E152">A511</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">n1</font>,<font color="Maroon" title="c55">n2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E153">A512</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">n1</font>,<font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E154">A513</font></span>: 
<font color="Maroon" title="c54">n1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E152">A511</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="lab"><font color="Green" title="E155">A514</font></span>: 
<font color="Maroon" title="c55">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E152">A511</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="kw">then </span>
<font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E126">A417</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E151">A510</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E150">A509</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E149">A508</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A515</font></span>: 
<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="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></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="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E144">A503</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A357</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E150">A509</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E149">A508</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E152">A511</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E153">A512</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1"/>
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E156">A515</font></span></a>, <a class="ref" href="seqm_3.html#T82" title="SEQM_3:th.82">SEQM_3:82</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/><span class="lab"><font color="Green" title="E157">A516</font></span>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E158">A517</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E154">A513</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E153">A512</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E154">A513</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"><span class="lab"><font color="Green" title="E157">A516</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A518</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E154">A513</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"><span class="lab"><font color="Green" title="E158">A517</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E62:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E145">A504</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_1"><span class="lab"><font color="Green" title="E159">A518</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/><span class="lab"><font color="Green" title="E157">A519</font></span>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">Lma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A520</font></span>: 
<font color="Maroon" title="c55">n2</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="c26">Lma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E155">A514</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c43">La</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E153">A512</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E155">A514</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"><span class="lab"><font color="Green" title="E157">A519</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c26">Lma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"><span class="lab"><font color="Green" title="E158">A520</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><span class="lab"><font color="Green" title="E151">A510</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E64:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E146">A521</font></span>: 
<font color="Maroon" title="c46">pl</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E137">A483</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E65:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A522</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E67:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="lab"><font color="Green" title="E148">A523</font></span>: 
<font color="Maroon" title="c42">Li</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Li</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E68:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E149">A524</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A365</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c53">i2</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> <span class="kw"> such that </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E150">A525</font></span>: 
<font color="Maroon" title="c53">i2</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="c41">C</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E151">A526</font></span>: 
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c53">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E150">A525</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E152">A527</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c53">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E66:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E147">A522</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E150">A525</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E151">A526</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E153">A528</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c53">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E66:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E147">A522</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E150">A525</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">assume </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E154">A529</font></span>: 
<font color="Maroon" title="c45">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">consider </span><font color="Maroon" title="c54">n1</font>, <font color="Maroon" title="c55">n2</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> <span class="kw"> such that </span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E155">A530</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c54">n1</font>,<font color="Maroon" title="c55">n2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E156">A531</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c54">n1</font>,<font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E157">A532</font></span>: 
<font color="Maroon" title="c54">n1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E155">A530</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="lab"><font color="Green" title="E158">A533</font></span>: 
<font color="Maroon" title="c55">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E155">A530</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="kw">then </span>
<font color="Maroon" title="c53">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E68:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E149">A524</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E154">A529</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E152">A527</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E153">A528</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A534</font></span>: 
<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="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></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="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E144">A503</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E110">A357</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E152">A527</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E153">A528</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E155">A530</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E156">A531</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1"/>
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> ) or (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E159">A534</font></span></a>, <a class="ref" href="seqm_3.html#T82" title="SEQM_3:th.82">SEQM_3:82</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/><span class="lab"><font color="Green" title="E160">A535</font></span>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E161">A536</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c41">C</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c41">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c54">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E157">A532</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c41">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c54">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E156">A531</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E157">A532</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"><span class="lab"><font color="Green" title="E160">A535</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E162">A537</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E157">A532</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"><span class="lab"><font color="Green" title="E161">A536</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E62:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E145">A504</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_1"><span class="lab"><font color="Green" title="E162">A537</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/><span class="lab"><font color="Green" title="E160">A538</font></span>: 
(  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c55">n2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c54">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c42">Li</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A539</font></span>: 
<font color="Maroon" title="c55">n2</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="c42">Li</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E158">A533</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c42">Li</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Li</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c42">Li</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c55">n2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E156">A531</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E158">A533</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"><span class="lab"><font color="Green" title="E160">A538</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c39">lg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c42">Li</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"><span class="lab"><font color="Green" title="E161">A539</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E140">A486</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><span class="lab"><font color="Green" title="E154">A529</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E70:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E150">A540</font></span>: 
<font color="Maroon" title="c45">pf</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E124">A415</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1"/>
( <font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">pl</font> or <font color="Maroon" title="c46">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">pf</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E139">A485</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E151">A541</font></span>: 
<font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c46">pl</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<font color="Maroon" title="c46">pl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c53">LL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c46">pl</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c45">pf</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E151">A541</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c54">w1</font> = <font color="Maroon" title="c45">pf</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A361</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c55">F1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>11</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <span class="kw">means </span>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E152">A542</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c56">F</font> being   <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><span class="kw"> such that </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E153">A543</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">LL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S1" title="FINSEQ_4:sch.1">FINSEQ_4:sch 1</a>(<a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E152">A542</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c57">hf</font> = <font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font>;<br/><span class="kw">set </span><font color="Maroon" title="c58">FF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E154">A544</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E155">A545</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c45">pf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A546</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c54">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c59">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> &amp; <font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="lab"><font color="Green" title="E157">A547</font></span>: 
<font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> &amp; <font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )</span><br/>

<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
<font color="Maroon" title="c59">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><span class="lab"><font color="Green" title="E157">A547</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c59">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">pf</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A548</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c59">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c59">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><span class="lab"><font color="Green" title="E157">A547</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><span class="lab"><font color="Green" title="E156">A546</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c46">pl</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><span class="lab"><font color="Green" title="E158">A548</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> &amp; <font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E156">A549</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c59">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font> or <font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c60">n</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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><span class="lab"><font color="Green" title="E157">A550</font></span>: 
<font color="Maroon" title="c60">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="c56">F</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><span class="lab"><font color="Green" title="E158">A551</font></span>: 
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="lab"><font color="Green" title="E157">A550</font></span></a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E159">A552</font></span>: 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="lab"><font color="Green" title="E157">A550</font></span></a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="lab"><font color="Green" title="E157">A550</font></span></a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="lab"><font color="Green" title="E158">A551</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><span class="lab"><font color="Green" title="E159">A552</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<font color="Maroon" title="c45">pf</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> <font color="Maroon" title="c46">pl</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E151">A541</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c45">pf</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> <font color="Maroon" title="c46">pl</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A553</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><div><span class="lab"><font color="Green" title="E158">A554</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c59">i1</font>, <font color="Maroon" title="c60">i2</font>, <font color="Maroon" title="c61">j1</font>, <font color="Maroon" title="c62">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font> &amp; <font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> &amp; 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="c52">h2</font> implies <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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E159">A555</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E160">A556</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E161">A557</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E162">A558</font></span>: 
<font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E163">A559</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> <font color="Maroon" title="c52">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E163">A559</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A560</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E165">A561</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="lab"><font color="Green" title="E166">A562</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</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="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c56">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="c56">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E166">A562</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E166">A562</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E126">A417</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E166">A562</font></span></a></span>
;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E167">A563</font></span>: 
( <font color="Maroon" title="c59">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> &amp; <font color="Maroon" title="c60">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E159">A555</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E161">A557</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E165">A561</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
<font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E163">A559</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c61">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c62">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E160">A556</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E162">A558</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E164">A560</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> = 
<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="c16">ma</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><span class="lab"><font color="Green" title="E167">A563</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E159">A564</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<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="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E160">A565</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><span class="lab"><font color="Green" title="E161">A566</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c47">K2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><span class="lab"><font color="Green" title="E162">A567</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c59">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c59">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><span class="lab"><font color="Green" title="E163">A568</font></span>: 
<font color="Maroon" title="c59">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><span class="lab"><font color="Green" title="E164">A569</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></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><br/>

<span class="kw">set </span><font color="Maroon" title="c60">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font>;<br/>
<span class="kw">assume </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><span class="lab"><font color="Green" title="E165">A570</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c61">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E159">A564</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E166">A571</font></span>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E164">A569</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1"/>
( <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E165">A570</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E166">A571</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E167">A572</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c62">i</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E168">A573</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E169">A574</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E170">A575</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E168">A573</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E169">A574</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">y</font> = <font color="Maroon" title="c61">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E169">A574</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E171">A576</font></span>: 
<font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E172">A577</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E167">A572</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E173">A578</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">CK1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E162">A567</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A579</font></span>: 
<font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E175">A580</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">CK1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E168">A573</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E173">A578</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/>
<font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E168">A573</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E173">A578</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E170">A575</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E173">A578</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E175">A580</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">CK1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E175">A580</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A581</font></span>: 
<font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E163">A568</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E172">A577</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="lab"><font color="Green" title="E177">A582</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c56">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</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"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c56">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="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A365</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E171">A576</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E174">A579</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A583</font></span>: 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E173">A578</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E176">A581</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E177">A582</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E164">A569</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_1"><span class="lab"><font color="Green" title="E178">A583</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E167">A584</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c62">i</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E168">A585</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E169">A586</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E170">A587</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E168">A585</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E169">A586</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">y</font> = <font color="Maroon" title="c61">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E169">A586</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E171">A588</font></span>: 
<font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E172">A589</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E167">A584</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E173">A590</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c48">CK2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E161">A566</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A591</font></span>: 
<font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E175">A592</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c48">CK2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E168">A585</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E173">A590</font></span></a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/>
<font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E168">A585</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E173">A590</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E170">A587</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E173">A590</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E175">A592</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c48">CK2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E175">A592</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A593</font></span>: 
<font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E163">A568</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E172">A589</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="lab"><font color="Green" title="E177">A594</font></span>: 
<font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A363</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E171">A588</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E174">A591</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E178">A595</font></span>: 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E173">A590</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E176">A593</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E177">A594</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><span class="lab"><font color="Green" title="E164">A569</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5_2_1_2"><span class="lab"><font color="Green" title="E178">A595</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c59">w</font> = <font color="Maroon" title="c54">w1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c60">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="lab"><font color="Green" title="E161">A596</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E162">A597</font></span>: 
<font color="Maroon" title="c60">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
<font color="Maroon" title="c60">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E161">A596</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">i1</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> <span class="kw"> such that </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="lab"><font color="Green" title="E163">A598</font></span>: 
<font color="Maroon" title="c61">i1</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="c56">F</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="lab"><font color="Green" title="E164">A599</font></span>: 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c60">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="lab"><font color="Green" title="E165">A600</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c56">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c61">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E163">A598</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">A601</font></span>: 
<font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="lab"><font color="Green" title="E167">A602</font></span>: 
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E163">A598</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E165">A600</font></span></a></span>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A603</font></span>: 
<font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</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="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
( <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E163">A598</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E165">A600</font></span></a></span>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E162">A597</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E164">A599</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E167">A602</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c61">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E168">A603</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E64:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E146">A521</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><span class="lab"><font color="Green" title="E166">A601</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></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="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E160">A565</font></span></a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A604</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c59">w</font> = <font color="Maroon" title="c54">w1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">let </span><font color="Maroon" title="c60">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c60">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="c56">F</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"/><span class="lab"><font color="Green" title="E162">A605</font></span>: 
<font color="Maroon" title="c60">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="c56">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#M2" title="SUBSET_1:mode.2">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">i</font>, <font color="Maroon" title="c62">j</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> <span class="kw"> such that </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"/><span class="lab"><font color="Green" title="E163">A606</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">i</font>,<font color="Maroon" title="c62">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">w</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">i</font>,<font color="Maroon" title="c62">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c63">i</font> = <font color="Maroon" title="c61">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">i</font>,<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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c63">i</font>,<font color="Olive" title="b1">j</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c64">j</font> = <font color="Maroon" title="c62">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">i</font>,<font color="Maroon" title="c64">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c63">i</font>,<font color="Maroon" title="c64">j</font> )</span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c60">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><span class="lab"><font color="Green" title="E162">A605</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c63">i</font>,<font color="Maroon" title="c64">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c63">i</font>,<font color="Maroon" title="c64">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><span class="lab"><font color="Green" title="E162">A605</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><span class="lab"><font color="Green" title="E163">A606</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A403</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E162">A607</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E134">A470</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><div><span class="lab"><font color="Green" title="E163">A608</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="kw">now </span></a><div class="add"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E164">A609</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c59">i1</font>, <font color="Maroon" title="c60">i2</font>, <font color="Maroon" title="c61">j1</font>, <font color="Maroon" title="c62">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font> &amp; <font color="Maroon" title="c56">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</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="c51">h1</font> &amp; 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="c56">F</font> implies <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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</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="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E165">A610</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E166">A611</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E167">A612</font></span>: 
<font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E168">A613</font></span>: 
<font color="Maroon" title="c56">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E169">A614</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</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="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="lab"><font color="Green" title="E170">A615</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> <font color="Maroon" title="c56">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/>
<font color="Maroon" title="c56">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E170">A615</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E170">A615</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A365</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E67:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E148">A523</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A616</font></span>: 
( <font color="Maroon" title="c61">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> &amp; <font color="Maroon" title="c62">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E166">A611</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E168">A613</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E164">A609</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/>
<font color="Maroon" title="c19">l1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E169">A614</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A617</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/>
<font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E169">A614</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c59">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l1</font> &amp; <font color="Maroon" title="c60">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E165">A610</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E167">A612</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E172">A617</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">mi</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_8"><span class="lab"><font color="Green" title="E171">A616</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c59">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c59">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="c56">F</font> &amp; <font color="Maroon" title="c59">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="c56">F</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/><span class="lab"><font color="Green" title="E164">A618</font></span>: 
<font color="Maroon" title="c59">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="c56">F</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/><span class="lab"><font color="Green" title="E165">A619</font></span>: 
<font color="Maroon" title="c59">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="c56">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/><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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E164">A618</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E166">A620</font></span>: 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E164">A618</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/>
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c59">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E165">A619</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E167">A621</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E165">A619</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c60">i1</font>, <font color="Maroon" title="c61">i2</font>, <font color="Maroon" title="c62">j1</font>, <font color="Maroon" title="c63">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</font> implies <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="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/><span class="lab"><font color="Green" title="E168">A622</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</font> &amp; <font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/>
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E155">A545</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E164">A618</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"/>
<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="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</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="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E166">A620</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E167">A621</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><span class="lab"><font color="Green" title="E168">A622</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><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" title="b2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">j2</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="#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A405</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E163">A608</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><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" title="b2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">j2</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="#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E135">A472</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E158">A554</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E162">A607</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A623</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E161">A604</font></span></a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">GOBOARD1:41</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c59">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A624</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E166">A625</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/><span class="lab"><font color="Green" title="E167">A626</font></span>: 
<font color="Maroon" title="c15">mi</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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/><span class="lab"><font color="Green" title="E168">A627</font></span>: 
<font color="Maroon" title="c45">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c46">pl</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A361</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E151">A541</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/><span class="lab"><font color="Green" title="E169">A628</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/>
<font color="Maroon" title="c51">h1</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="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="lab"><font color="Green" title="E167">A626</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c56">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c56">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="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="lab"><font color="Green" title="E169">A628</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="lab"><font color="Green" title="E169">A628</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="lab"><font color="Green" title="E168">A627</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_1"><span class="lab"><font color="Green" title="E167">A626</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="lab"><font color="Green" title="E167">A629</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E167">A629</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><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="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A630</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A631</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A632</font></span>: 
<font color="Maroon" title="c49">K1</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="c11">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A633</font></span>: 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c56">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E168">A630</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/><span class="kw">then </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">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</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#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E132">A465</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E169">A631</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E132">A465</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E169">A631</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E171">A633</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_10_1_2"><span class="lab"><font color="Green" title="E170">A632</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="lab"><font color="Green" title="E167">A634</font></span>: 
<font color="Maroon" title="c54">w1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">pl</font>
 ;<br/><div><span class="lab"><font color="Green" title="E168">A635</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1"/>
( <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/><span class="lab"><font color="Green" title="E169">A636</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A637</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c46">pl</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/><span class="lab"><font color="Green" title="E171">A638</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</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="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E157">A553</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/>
<font color="Maroon" title="c52">h2</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="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E169">A636</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A639</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c56">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E172">A639</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E171">A638</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E153">A543</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E154">A544</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E167">A634</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E171">A638</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E170">A637</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_1"><span class="lab"><font color="Green" title="E169">A636</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="lab"><font color="Green" title="E169">A640</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/>
<font color="Maroon" title="c16">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E169">A640</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A641</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A642</font></span>: 
<font color="Maroon" title="c20">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A643</font></span>: 
<font color="Maroon" title="c47">K2</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="c12">Ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A644</font></span>: 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/><span class="lab"><font color="Green" title="E174">A645</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</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="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E170">A641</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">l2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E174">A645</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">l2</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E50:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E136">A482</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E171">A642</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E173">A644</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_1_2"><span class="lab"><font color="Green" title="E172">A643</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A348</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A351</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A646</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A177</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E170">A647</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </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">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E165">A624</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A648</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E26:1_1"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E166">A625</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E161">A604</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><a NAME="E40:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E170">A647</font></span></a>, <a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E165">A624</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A649</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E27:1_1"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A152</font></span></a>, <a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E168">A635</font></span></a>, <a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E161">A604</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2"><span class="lab"><font color="Green" title="E51">A92</font></span></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">GOBOARD1:26</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A350</font></span></a>, <a class="txt" href="#E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E170">A647</font></span></a>, <a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E165">A624</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E164">A623</font></span></a>, <a class="txt" href="#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E171">A648</font></span></a>, <a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E172">A649</font></span></a></span>;<br/><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A650</font></span>: 
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>
 ;<br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="#E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E143">A489</font></span></a>, <a class="txt" href="#E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E159">A564</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E130">A421</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c56">F</font></span>)</span>

;<br/><a NAME="E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c56">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E169">A646</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E156">A549</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><span class="lab"><font color="Green" title="E173">A650</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E151">A651</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c45">pf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<font color="Maroon" title="c45">pf</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c53">LL</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">pl</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E151">A651</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c54">F1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>11</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <span class="kw">means </span>for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E152">A652</font></span>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">n</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c55">n</font>, <font color="Maroon" title="c56">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> &amp; <font color="Maroon" title="c56">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> implies ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> &amp; <font color="Maroon" title="c56">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="lab"><font color="Green" title="E153">A653</font></span>: 
<font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="lab"><font color="Green" title="E154">A654</font></span>: 
<font color="Maroon" title="c56">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> &amp; <font color="Maroon" title="c56">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="lab"><font color="Green" title="E155">A655</font></span>: 
<font color="Maroon" title="c55">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E153">A653</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c55">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E153">A653</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E156">A656</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c53">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c57">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E155">A655</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E155">A655</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E138">A484</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c57">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c45">pf</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E156">A656</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font> &amp; <font color="Maroon" title="c56">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><span class="lab"><font color="Green" title="E154">A654</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E153">A657</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">n</font> is    <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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c55">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c46">pl</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E154">A658</font></span>: 
<font color="Maroon" title="c53">LL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/>
<font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c55">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><span class="lab"><font color="Green" title="E154">A658</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E154">A659</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c55">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> implies  ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Maroon" title="c55">n</font>,<font color="Olive" title="b1">p</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/><span class="lab"><font color="Green" title="E155">A660</font></span>: 
<font color="Maroon" title="c55">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>11</sub></font>[<font color="Maroon" title="c55">n</font>,<font color="Olive" title="b1">p</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c56">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c55">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E153">A657</font></span></a></span>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>11</sub></font>[<font color="Maroon" title="c55">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/>
<font color="Maroon">S<sub>11</sub></font>[<font color="Maroon" title="c55">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c56">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"><span class="lab"><font color="Green" title="E155">A660</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c55">F</font> being   <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><span class="kw"> such that </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E155">A661</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">LL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S1" title="FINSEQ_4:sch.1">FINSEQ_4:sch 1</a>(<a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E154">A659</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c56">hf</font> = <font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font>;<br/><span class="kw">set </span><font color="Maroon" title="c57">FF</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E156">A662</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E157">A663</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c58">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font> or <font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/>
<font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c59">n</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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><span class="lab"><font color="Green" title="E158">A664</font></span>: 
<font color="Maroon" title="c59">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="c55">F</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><span class="lab"><font color="Green" title="E159">A665</font></span>: 
<font color="Maroon" title="c58">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c60">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c59">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E153">A657</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><span class="lab"><font color="Green" title="E158">A664</font></span></a></span>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><span class="lab"><font color="Green" title="E158">A664</font></span></a></span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">u</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><span class="lab"><font color="Green" title="E158">A664</font></span></a></span>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><span class="lab"><font color="Green" title="E159">A665</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/>
<font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1"><span class="lab"><font color="Green" title="E19">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<font color="Maroon" title="c46">pl</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> <font color="Maroon" title="c45">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E151">A651</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c46">pl</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> <font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E158">A666</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><div><span class="lab"><font color="Green" title="E159">A667</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c58">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">LL</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">let </span><font color="Maroon" title="c59">i1</font>, <font color="Maroon" title="c60">i2</font>, <font color="Maroon" title="c61">j1</font>, <font color="Maroon" title="c62">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font> &amp; <font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> &amp; 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="c52">h2</font> implies <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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</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="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E160">A668</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E161">A669</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E162">A670</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c59">i1</font>,<font color="Maroon" title="c60">i2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E163">A671</font></span>: 
<font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c61">j1</font>,<font color="Maroon" title="c62">j2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E164">A672</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> <font color="Maroon" title="c52">h2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E164">A672</font></span></a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A673</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E166">A674</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="lab"><font color="Green" title="E167">A675</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</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="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c55">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="c55">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E167">A675</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E167">A675</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E167">A675</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c16">ma</font>,<font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A363</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E125">A416</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A676</font></span>: 
( <font color="Maroon" title="c59">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> &amp; <font color="Maroon" title="c60">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E160">A668</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E162">A670</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E166">A674</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
<font color="Maroon" title="c52">h2</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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E164">A672</font></span></a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><span class="kw">then </span>
( <font color="Maroon" title="c61">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c62">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c47">K2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E161">A669</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E163">A671</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E165">A673</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c59">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c61">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c60">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j2</font></span>)</span></span>)</span> = 
<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="c16">ma</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><span class="lab"><font color="Green" title="E168">A676</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c58">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c58">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="c55">F</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><span class="lab"><font color="Green" title="E160">A677</font></span>: 
<font color="Maroon" title="c58">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="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#M2" title="SUBSET_1:mode.2">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c59">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c58">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E153">A657</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><span class="lab"><font color="Green" title="E161">A678</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="lab"><font color="Green" title="E160">A677</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="lab"><font color="Green" title="E160">A677</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c60">i</font>, <font color="Maroon" title="c61">j</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> <span class="kw"> such that </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><span class="lab"><font color="Green" title="E162">A679</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c60">i</font>,<font color="Maroon" title="c61">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c60">i</font>,<font color="Maroon" title="c61">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c62">i</font> = <font color="Maroon" title="c60">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">i</font>,<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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i</font>,<font color="Olive" title="b1">j</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c63">j</font> = <font color="Maroon" title="c61">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c63">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c63">j</font> )</span><br/><span class="kw">thus </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c63">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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c58">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c63">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="lab"><font color="Green" title="E160">A677</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="lab"><font color="Green" title="E161">A678</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><span class="lab"><font color="Green" title="E162">A679</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A403</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E160">A680</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <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="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E134">A470</font></span></a>, <a class="ref" href="goboard1.html#T39" title="GOBOARD1:th.39">GOBOARD1:39</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c58">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E161">A681</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="lab"><font color="Green" title="E162">A682</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<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="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E163">A683</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/><span class="lab"><font color="Green" title="E164">A684</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c47">K2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/><span class="lab"><font color="Green" title="E165">A685</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c49">K1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c59">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c59">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/><span class="lab"><font color="Green" title="E166">A686</font></span>: 
<font color="Maroon" title="c59">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/><span class="lab"><font color="Green" title="E167">A687</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></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><br/>

<span class="kw">set </span><font color="Maroon" title="c60">gg</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font>;<br/>
<span class="kw">assume </span><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/><span class="lab"><font color="Green" title="E168">A688</font></span>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c61">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E162">A682</font></span></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E169">A689</font></span>: <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E167">A687</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1"/>
( <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> or <font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E168">A688</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E169">A689</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E170">A690</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A691</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E172">A692</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">pf</font>
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A693</font></span>: 
( <font color="Maroon" title="c55">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E172">A692</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E170">A690</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c62">i</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> <span class="kw"> such that </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E174">A694</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E175">A695</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E176">A696</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E174">A694</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E175">A695</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">y</font> = <font color="Maroon" title="c61">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E175">A695</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="lab"><font color="Green" title="E177">A697</font></span>: 
<font color="Maroon" title="c17">Lmi</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Lmi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E178">A698</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">CK1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E165">A685</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A699</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c50">CK1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E174">A694</font></span></a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
<font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E174">A694</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E178">A698</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E176">A696</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E178">A698</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E179">A699</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c50">CK1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E179">A699</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E180">A700</font></span>: 
<font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E166">A686</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E171">A691</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
<font color="Maroon" title="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c50">CK1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E178">A698</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c55">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="c50">CK1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A365</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E177">A697</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E173">A693</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E181">A701</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E178">A698</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E180">A700</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E172">A692</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E167">A687</font></span></a>, <a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_1"><span class="lab"><font color="Green" title="E181">A701</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E170">A702</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c62">i</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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E171">A703</font></span>: 
<font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E172">A704</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c62">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E173">A705</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E171">A703</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E172">A704</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">y</font> = <font color="Maroon" title="c61">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E172">A704</font></span></a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E174">A706</font></span>: 
<font color="Maroon" title="c61">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E170">A702</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c64">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c53">LL</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E175">A707</font></span>: 
<font color="Maroon" title="c26">Lma</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">Lma</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="lab"><font color="Green" title="E176">A708</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c48">CK2</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E164">A684</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A709</font></span>: 
<font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E178">A710</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c48">CK2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E171">A703</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E176">A708</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/>
<font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E133">A466</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E171">A703</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E176">A708</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c62">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E173">A705</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E176">A708</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E178">A710</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c63">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c48">CK2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E178">A710</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E179">A711</font></span>: 
<font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E166">A686</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E174">A706</font></span></a>, <a class="ref" href="goboard1.html#T20" title="GOBOARD1:th.20">GOBOARD1:20</a></span>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="lab"><font color="Green" title="E180">A712</font></span>: 
<font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c64">u</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c64">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c64">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a></span>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c48">CK2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A363</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E175">A707</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E177">A709</font></span></a>, <a class="ref" href="goboard1.html#T17" title="GOBOARD1:th.17">GOBOARD1:17</a></span>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E181">A713</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c59">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E176">A708</font></span></a>, <a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E179">A711</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E180">A712</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/><span class="kw">hence </span><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><span class="lab"><font color="Green" title="E167">A687</font></span></a>, <a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_2_1_2"><span class="lab"><font color="Green" title="E181">A713</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c59">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="lab"><font color="Green" title="E164">A714</font></span>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E165">A715</font></span>: 
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/>
<font color="Maroon" title="c59">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E164">A714</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c60">i1</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> <span class="kw"> such that </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="lab"><font color="Green" title="E166">A716</font></span>: 
<font color="Maroon" title="c60">i1</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="c55">F</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="lab"><font color="Green" title="E167">A717</font></span>: 
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c59">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c61">w</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c60">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E153">A657</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E166">A716</font></span></a></span>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c60">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E166">A716</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A718</font></span>: 
<font color="Maroon" title="c61">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="lab"><font color="Green" title="E169">A719</font></span>: 
<font color="Maroon" title="c61">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E166">A716</font></span></a></span>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A720</font></span>: 
<font color="Maroon" title="c61">w</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="c7">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/>
( <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E166">A716</font></span></a></span>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c41">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E23:1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E165">A715</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E167">A717</font></span></a>, <a class="txt" href="#E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E169">A719</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c61">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E170">A720</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1"><span class="lab"><font color="Green" title="E18">A17</font></span></a>, <a class="txt" href="#E70:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E150">A540</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_9"><span class="lab"><font color="Green" title="E168">A718</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></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="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E163">A683</font></span></a></span>;<br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E164">A721</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><span class="lab"><font color="Green" title="E165">A722</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c59">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c59">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="c55">F</font> &amp; <font color="Maroon" title="c59">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="c55">F</font> implies for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/>1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/><span class="lab"><font color="Green" title="E166">A723</font></span>: 
<font color="Maroon" title="c59">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="c55">F</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/><span class="lab"><font color="Green" title="E167">A724</font></span>: 
<font color="Maroon" title="c59">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="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font> holds <br/>1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="b1">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b2">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c60">w3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c59">n</font>, <font color="Maroon" title="c61">w2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E153">A657</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E166">A723</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E167">A724</font></span></a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">w3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E166">A723</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A725</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c60">w3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E166">A723</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/>
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E167">A724</font></span></a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A726</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c61">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E167">A724</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c62">i1</font>, <font color="Maroon" title="c63">i2</font>, <font color="Maroon" title="c64">j1</font>, <font color="Maroon" title="c65">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">i1</font>,<font color="Maroon" title="c63">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c64">j1</font>,<font color="Maroon" title="c65">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i1</font>,<font color="Maroon" title="c63">i2</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c64">j1</font>,<font color="Maroon" title="c65">j2</font> implies 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c62">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c64">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c63">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c65">j2</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/><span class="lab"><font color="Green" title="E170">A727</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">i1</font>,<font color="Maroon" title="c63">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c64">j1</font>,<font color="Maroon" title="c65">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">i1</font>,<font color="Maroon" title="c63">i2</font> &amp; <font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c59">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c64">j1</font>,<font color="Maroon" title="c65">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c62">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c64">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c63">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c65">j2</font></span>)</span></span>)</span></span><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"/>
<font color="Maroon" title="c61">w2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c59">n</font>
 ;<br/><span class="kw">hence </span>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="c64">j1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">i1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c65">j2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">i2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1"><span class="lab"><font color="Green" title="E16">A15</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E168">A725</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E169">A726</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><span class="lab"><font color="Green" title="E170">A727</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c62">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c64">j1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c63">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c65">j2</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<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="c62">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c64">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c63">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c65">j2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
<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="c62">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c64">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c63">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c65">j2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="lab"><font color="Green" title="E166">A728</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c45">pf</font>
 ;<br/><div><span class="lab"><font color="Green" title="E167">A729</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/><span class="lab"><font color="Green" title="E168">A730</font></span>: 
<font color="Maroon" title="c15">mi</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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/><span class="lab"><font color="Green" title="E169">A731</font></span>: 
<font color="Maroon" title="c45">pf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c45">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E113">A361</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/><span class="lab"><font color="Green" title="E170">A732</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c53">LL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/>
<font color="Maroon" title="c51">h1</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="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="lab"><font color="Green" title="E168">A730</font></span></a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c55">F</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c55">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="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="lab"><font color="Green" title="E170">A732</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E166">A728</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="lab"><font color="Green" title="E170">A732</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c45">pf</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="lab"><font color="Green" title="E169">A731</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_1"><span class="lab"><font color="Green" title="E168">A730</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="lab"><font color="Green" title="E168">A733</font></span>: 
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E168">A733</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><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="c15">mi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A734</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E170">A735</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A736</font></span>: 
<font color="Maroon" title="c49">K1</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="c11">L1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E172">A737</font></span>: 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c55">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E169">A734</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/><span class="kw">then </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">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</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#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c51">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E132">A465</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E170">A735</font></span></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E132">A465</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E170">A735</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">L1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E172">A737</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_11_1_2"><span class="lab"><font color="Green" title="E171">A736</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c33">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A348</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A351</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E168">A738</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A177</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="kw">now </span></a><div class="add"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E169">A739</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:1_1"><span class="lab"><font color="Green" title="E32">A39</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c59">w4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">let </span><font color="Maroon" title="c60">i1</font>, <font color="Maroon" title="c61">i2</font>, <font color="Maroon" title="c62">j1</font>, <font color="Maroon" title="c63">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</font> &amp; <font color="Maroon" title="c55">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</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="c51">h1</font> &amp; 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="c55">F</font> implies <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="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E170">A740</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E171">A741</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E172">A742</font></span>: 
<font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c60">i1</font>,<font color="Maroon" title="c61">i2</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E173">A743</font></span>: 
<font color="Maroon" title="c55">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="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c62">j1</font>,<font color="Maroon" title="c63">j2</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E174">A744</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</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="c51">h1</font>
 <span class="kw">and </span><br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="lab"><font color="Green" title="E175">A745</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> <font color="Maroon" title="c55">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">w4</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E175">A745</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c59">w4</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E152">A652</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E175">A745</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">mi</font>,<font color="Maroon" title="c49">K1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A365</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="txt" href="#E67:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E148">A523</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A746</font></span>: 
( <font color="Maroon" title="c62">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> &amp; <font color="Maroon" title="c63">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E171">A741</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E173">A743</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E169">A739</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
<font color="Maroon" title="c19">l1</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A398</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E174">A744</font></span></a></span>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A747</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c49">K1</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> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A95</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A364</font></span></a>, <a class="txt" href="#E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A397</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
<font color="Maroon" title="c51">h1</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="c51">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c19">l1</font>,<font color="Maroon" title="c49">K1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A366</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E174">A744</font></span></a></span>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><span class="kw">then </span>
( <font color="Maroon" title="c60">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l1</font> &amp; <font color="Maroon" title="c61">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c49">K1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E170">A740</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E172">A742</font></span></a>, <a class="txt" href="#E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E177">A747</font></span></a>, <a class="ref" href="goboard1.html#T21" title="GOBOARD1:th.21">GOBOARD1:21</a></span>;<br/><span class="kw">hence </span><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="c60">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c62">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c61">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c63">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">mi</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">mi</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><span class="lab"><font color="Green" title="E176">A746</font></span></a>, <a class="ref" href="absvalue.html#T7" title="ABSVALUE:th.7">ABSVALUE:7</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> holds <br/>for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font> holds <br/><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" title="b2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">i2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">j2</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="#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A405</font></span></a>, <a class="txt" href="#E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E165">A722</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> &amp; <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">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> <font color="Maroon" title="c5">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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> <font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font> holds <br/><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" title="b2">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="b3">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b5">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="#E49:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E135">A472</font></span></a>, <a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E159">A667</font></span></a>, <a class="ref" href="goboard1.html#T40" title="GOBOARD1:th.40">GOBOARD1:40</a></span>;<br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E160">A680</font></span></a>, <a class="ref" href="goboard1.html#D11" title="GOBOARD1:def.11">GOBOARD1:def 11</a></span>;<br/><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E169">A748</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E164">A721</font></span></a>, <a class="ref" href="goboard1.html#T41" title="GOBOARD1:th.41">GOBOARD1:41</a></span>;<br/><span class="lab"><font color="Green" title="E170">A749</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </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">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E161">A681</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E171">A750</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E26:1_1"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E167">A729</font></span></a>, <a class="txt" href="#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E164">A721</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><div><span class="lab"><font color="Green" title="E172">A751</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1"/>
( <font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> or <font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/><span class="lab"><font color="Green" title="E173">A752</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/><span class="lab"><font color="Green" title="E174">A753</font></span>: 
<font color="Maroon" title="c46">pl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c45">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E138">A484</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E151">A651</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/><span class="lab"><font color="Green" title="E175">A754</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c45">pf</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c46">pl</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">pl</font>
 ;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/><span class="lab"><font color="Green" title="E176">A755</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</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="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E158">A666</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/>
<font color="Maroon" title="c52">h2</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="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E173">A752</font></span></a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A756</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c55">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c53">LL</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E177">A756</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E176">A755</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c45">pf</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E155">A661</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E156">A662</font></span></a>, <a class="txt" href="#E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E175">A754</font></span></a>, <a class="txt" href="#E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E176">A755</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">g2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c46">pl</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A356</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E174">A753</font></span></a>, <a class="ref" href="finseq_4.html#T86" title="FINSEQ_4:th.86">FINSEQ_4:86</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E112">A360</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_1"><span class="lab"><font color="Green" title="E173">A752</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="lab"><font color="Green" title="E173">A757</font></span>: 
<font color="Maroon" title="c16">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/>
<font color="Maroon" title="c16">ma</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1"><span class="lab"><font color="Green" title="E34">A41</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E173">A757</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">ma</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A758</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E175">A759</font></span>: 
<font color="Maroon" title="c20">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c20">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E176">A760</font></span>: 
<font color="Maroon" title="c47">K2</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="c12">Ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E177">A761</font></span>: 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c47">K2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/><span class="lab"><font color="Green" title="E178">A762</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</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="c52">h2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E174">A758</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
<font color="Maroon" title="c52">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c20">l2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E178">A762</font></span></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></span>
<br/>.= 
<font color="Maroon" title="c5">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ma</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">l2</font></span>)</span>,<font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E127">A418</font></span></a>, <a class="txt" href="#E50:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E136">A482</font></span></a>, <a class="txt" href="#E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E175">A759</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">Ll</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c47">K2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E114">A362</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A352</font></span></a>, <a class="txt" href="#E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E177">A761</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14_1_2"><span class="lab"><font color="Green" title="E176">A760</font></span></a>, <a class="ref" href="partfun2.html#T4" title="PARTFUN2:th.4">PARTFUN2:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E40:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E170">A749</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E161">A681</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E173">A763</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span> <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E27:1_1"><span class="lab"><font color="Green" title="E24">A27</font></span></a>, <a class="txt" href="#E27:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A151</font></span></a>, <a class="txt" href="#E28:1_1_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A152</font></span></a>, <a class="txt" href="#E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E172">A751</font></span></a>, <a class="txt" href="#E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E164">A721</font></span></a>, <a class="ref" href="goboard1.html#T37" title="GOBOARD1:th.37">GOBOARD1:37</a></span>;<br/><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">DelCol</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E63:1_1"><span class="lab"><font color="Green" title="E48">A74</font></span></a>, <a class="txt" href="#E1:1_1_5_1_2"><span class="lab"><font color="Green" title="E51">A92</font></span></a>, <a class="ref" href="goboard1.html#T26" title="GOBOARD1:th.26">GOBOARD1:26</a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1_5_1_2"><span class="lab"><font color="Green" title="E52">A93</font></span></a>, <a class="txt" href="#E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A350</font></span></a>, <a class="txt" href="#E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E170">A749</font></span></a>, <a class="txt" href="#E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E161">A681</font></span></a>, <a class="txt" href="#E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E169">A748</font></span></a>, <a class="txt" href="#E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E171">A750</font></span></a>, <a class="txt" href="#E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E173">A763</font></span></a></span>;<br/><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E174">A764</font></span>: 
<font color="Maroon" title="c58">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span>
 ;<br/><span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c51">h1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="#E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E143">A489</font></span></a>, <a class="txt" href="#E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E162">A682</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"><span class="lab"><font color="Green" title="E130">A421</font></span></a>, <a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c55">F</font></span>)</span>

;<br/><a NAME="E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c51">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c55">F</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c52">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c38">tt</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E168">A738</font></span></a>, <a class="txt" href="#E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E157">A663</font></span></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><span class="lab"><font color="Green" title="E174">A764</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E72:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E69:1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:1"/><span class="lab"><font color="Green" title="E3">A765</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:1"/><span class="lab"><font color="Green" title="E4">A766</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A765</font></span></a>, <a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E5">A767</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G1</font> being  <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a><br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being  <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>  st <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="Olive" title="b2">g1</font> &amp; 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="Olive" title="b3">g2</font> &amp; <font color="Olive" title="b2">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b3">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b2">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">g1</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="Olive" title="b2">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b3">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b3">g2</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="Olive" title="b3">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Olive" title="b1">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G1</font></span>)</span></span>)</span> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <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>  st <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="Olive" title="b1">g1</font> &amp; 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="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b1">g1</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="Olive" title="b1">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">g2</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="Olive" title="b2">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">g2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="c6">g1</font> &amp; 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="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c6">g1</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="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c7">g2</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="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:1_2"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 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="c6">g1</font> &amp; 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="c7">g2</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c6">g1</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="c6">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,1</span>)</span> &amp; <font color="Maroon" title="c7">g2</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="c7">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G1</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span><br/><a NAME="E2:1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E4">A766</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:1"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</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="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/>
<a NAME="E6:1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A768</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:1"/>
( 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="c2">f1</font> &amp; 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">f2</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">f2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c2">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c2">f1</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="c2">f1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,1</span>)</span> &amp; <font color="Maroon" title="c3">f2</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">f2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">G</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f2</font></span><br/>

<span class="kw">hence </span><a NAME="E8:1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A767</font></span></a>, <a class="txt" href="#E6:1"><span class="lab"><font color="Green" title="E6">A768</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
