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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c2">f</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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E38">A1</font></i>: 
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="c2">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c1">G</font> &amp; <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</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> )
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E39">A2</font></i>: 
( not <font color="Maroon" title="c2">f</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> )
 <b>and </b><br/><a NAME="E3:41"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>and </b><br/><a NAME="E5:41"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 4
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">f'</font> = <font color="Maroon" title="c2">f</font> as  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a> <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topreal8.html#E2:41"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="topreal8.html#E3:41"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="topreal8.html#E4:41"><i><font color="Green" title="E41">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">f'</font> = <font color="Maroon" title="c3">f'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a> <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:41"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E2:41"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="topreal8.html#T3" target="_self" title="TOPREAL8:th.3">Th3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">h1</font> = <font color="Maroon" title="c4">f'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2;<br/>
<b>set </b><font color="Maroon" title="c6">h2</font> = <font color="Maroon" title="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1;<br/>
<a NAME="E9:41"/><i><font color="Green" title="E44">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c4">f'</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 2 as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/>
<a NAME="E11:41"/>
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">f</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:41"/><b>then </b><i><font color="Green" title="E45">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E13:41"/><b>then </b><i><font color="Green" title="E46">A9</font></i>: 
<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="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c2">f</font>
 
;<br/>
<a NAME="E14:41"/><i><font color="Green" title="E47">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green" title="E43">A6</font></i></a>, <a class="txt" href="topreal8.html#E12:41"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E15:41"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">h2</font> = <font color="Maroon" title="c4">f'</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/>
<a NAME="E17:41"/><i><font color="Green" title="E49">A12</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c8">h2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">h2</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1
 
<b>by </b><i><a class="ref" href="topreal8.html#T13" target="_self" title="TOPREAL8:th.13">Th13</a></i>;<br/>
<i><font color="Green" title="E50">A13</font></i>: <font color="Maroon" title="c2">f</font> = 
<font color="Maroon" title="c7">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>
<br/>.= 
<font color="Maroon" title="c7">h1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default">2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="graph_2.html#K2" title="GRAPH_2:func.2">-cut</a> <font color="Maroon" title="c8">h2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E17:41"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="finseq_6.html#T87" title="FINSEQ_6:th.87">FINSEQ_6:87</a></i>
<br/>.= 
<font color="Maroon" title="c7">h1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c8">h2</font>
<b>by </b><i><a class="ref" href="graph_2.html#D2" title="GRAPH_2:def.2">GRAPH_2:def 2</a></i>
;<br/>
<a NAME="E19:41"/><i><font color="Green" title="E51">A14</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">h1</font> <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="c2">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T20" title="FINSEQ_5:th.20">FINSEQ_5:20</a></i>;<br/>
<a NAME="E20:41"/><i><font color="Green" title="E52">A15</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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">h1</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E53">A16</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">h1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c11">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green" title="E54">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">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="c1">G</font>
 <b>and </b><br/><a NAME="E4:41_2"/><i><font color="Green" title="E55">A18</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="topreal8.html#E19:41"><i><font color="Green" title="E51">A14</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c11">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c13">j</font>
;<br/>

<b>thus </b><a NAME="E6:41_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">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="c1">G</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:41_2"><i><font color="Green" title="E54">A17</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:41_2"/>
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="txt" href="topreal8.html#E4:41_2"><i><font color="Green" title="E55">A18</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:41"/><i><font color="Green" title="E53">A19</font></i>: 
<font color="Maroon" title="c7">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="topreal8.html#E8:41"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="topreal8.html#T22" target="_self" title="TOPREAL8:th.22">Th22</a></i>;<br/>
<a NAME="E22:41"/>
<font color="Maroon" title="c7">h1</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E8:41"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="topreal8.html#T20" target="_self" title="TOPREAL8:th.20">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">g1</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><b> such that </b><br/><a NAME="E24:41"/><i><font color="Green" title="E54">A20</font></i>: 
<font color="Maroon" title="c9">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E25:41"/><i><font color="Green" title="E55">A21</font></i>: 
( <font color="Maroon" title="c9">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c9">g1</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c9">g1</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c9">g1</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> )
 <b>and </b><br/><a NAME="E26:41"/><i><font color="Green" title="E56">A22</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c9">g1</font>
 <b>and </b><br/><a NAME="E27:41"/><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 <b>and </b><br/><a NAME="E28:41"/><i><font color="Green" title="E58">A24</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">g1</font></span>)</span>
 <b>and </b><br/><a NAME="E29:41"/><i><font color="Green" title="E59">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h1</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="c9">g1</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E20:41"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="topreal8.html#E21:41"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="goboard3.html#T1" title="GOBOARD3:th.1">GOBOARD3:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">g1</font> = <font color="Maroon" title="c9">g1</font> as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/>
<a NAME="E31:41"/><i><font color="Green" title="E60">A26</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c8">h2</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_3"/><i><font color="Green" title="E61">A27</font></i>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">h2</font>
 ;<br/>

<a NAME="E2:41_3"/><b>then </b>
<font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c2">f</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T29" title="FINSEQ_5:th.29">FINSEQ_5:29</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c13">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:41_3"/><i><font color="Green" title="E62">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">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="c1">G</font>
 <b>and </b><br/><a NAME="E5:41_3"/><i><font color="Green" title="E63">A29</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c12">i</font>, <font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c13">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">i</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c15">j</font>
;<br/>

<b>thus </b><a NAME="E7:41_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:41_3"><i><font color="Green" title="E62">A28</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:41_3"/>
<font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:41_3"><i><font color="Green" title="E61">A27</font></i></a>, <a class="txt" href="topreal8.html#E5:41_3"><i><font color="Green" title="E63">A29</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:41"/><i><font color="Green" title="E61">A30</font></i>: 
<font color="Maroon" title="c8">h2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="topreal8.html#T24" target="_self" title="TOPREAL8:th.24">Th24</a></i>;<br/>
<a NAME="E33:41"/>
<font color="Maroon" title="c8">h2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>
 
<b>by </b><i><a class="ref" href="topreal8.html#T21" target="_self" title="TOPREAL8:th.21">Th21</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">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><b> such that </b><br/><a NAME="E35:41"/><i><font color="Green" title="E62">A31</font></i>: 
<font color="Maroon" title="c11">g2</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font>
 <b>and </b><br/><a NAME="E36:41"/><i><font color="Green" title="E63">A32</font></i>: 
( <font color="Maroon" title="c11">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c11">g2</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c11">g2</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c11">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> )
 <b>and </b><br/><a NAME="E37:41"/><i><font color="Green" title="E64">A33</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c11">g2</font>
 <b>and </b><br/><a NAME="E38:41"/><i><font color="Green" title="E65">A34</font></i>: 
<font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 <b>and </b><br/><a NAME="E39:41"/><i><font color="Green" title="E66">A35</font></i>: 
<font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font></span>)</span>
 <b>and </b><br/><a NAME="E40:41"/><i><font color="Green" title="E67">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</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="c11">g2</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E31:41"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="topreal8.html#E32:41"><i><font color="Green" title="E61">A30</font></i></a>, <a class="ref" href="goboard3.html#T1" title="GOBOARD3:th.1">GOBOARD3:1</a></i>;<br/>
<a NAME="E41:41"/><i><font color="Green" title="E68">A37</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">g2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E15:41"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="topreal8.html#E40:41"><i><font color="Green" title="E67">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">g2</font> = <font color="Maroon" title="c11">g2</font> as  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="realset1.html#T13" title="REALSET1:th.13">REALSET1:13</a></i>;<br/>
<b>take </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c10">g1</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c12">g2</font>;<br/>

<a NAME="E43:41"/><i><font color="Green" title="E69">A38</font></i>: 
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="c8">h2</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E14:41"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E44:41"/><i><font color="Green" title="E70">A39</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">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="c8">h2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E45:41"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">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="c7">h1</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E46:41"/><b>then </b><i><font color="Green" title="E71">A40</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>
<a NAME="E47:41"/><b>then </b><i><font color="Green" title="E72">A41</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</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">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E43:41"><i><font color="Green" title="E69">A38</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E48:41"/><i><font color="Green" title="E73">A42</font></i>: 
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">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E49:41"/><b>then </b><i><font color="Green" title="E74">A43</font></i>: 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>;<br/>
<b>then </b><i><font color="Green" title="E75">A44</font></i>: <font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E13:41"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="topreal8.html#E44:41"><i><font color="Green" title="E70">A39</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
;<br/>
<b>thus </b><a NAME="E51:41"/>
<font color="Maroon" title="c13">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E24:41"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E35:41"><i><font color="Green" title="E62">A31</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="topreal8.html#T12" target="_self" title="TOPREAL8:th.12">Th12</a></i>;<br/>

<a NAME="E52:41"/>
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="c12">g2</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E41:41"><i><font color="Green" title="E68">A37</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:41"/><b>then </b><i><font color="Green" title="E76">A45</font></i>: 
<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="c12">g2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c12">g2</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E54:41"/>
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="c10">g1</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E55:41"/><b>then </b><i><font color="Green" title="E77">A46</font></i>: 
<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="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c10">g1</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E56:41"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="topreal8.html#E12:41"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E57:41"/><b>then </b>
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">g2</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E40:41"><i><font color="Green" title="E67">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E58:41"/><b>then </b><i><font color="Green" title="E78">A47</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c12">g2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T70" title="BINARITH:th.70">BINARITH:70</a></i>;<br/>
<a NAME="E59:41"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:41"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="topreal8.html#E46:41"><i><font color="Green" title="E71">A40</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="topreal8.html#E49:41"><i><font color="Green" title="E74">A43</font></i></a>, <a class="ref" href="topreal8.html#T27" target="_self" title="TOPREAL8:th.27">Th27</a></i>;<br/>
<a NAME="E60:41"/><b>then </b><i><font color="Green" title="E79">A48</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E26:41"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="topreal8.html#E27:41"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="topreal8.html#E37:41"><i><font color="Green" title="E64">A33</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="topreal8.html#T19" target="_self" title="TOPREAL8:th.19">Th19</a></i>;<br/>
<a NAME="E61:41"/>
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="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E9:41"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="topreal8.html#E29:41"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="binarith.html#T73" title="BINARITH:th.73">BINARITH:73</a></i>;<br/>
<a NAME="E62:41"/><b>then </b><i><font color="Green" title="E80">A49</font></i>: 
<font color="Maroon" title="c10">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<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="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E55:41"><i><font color="Green" title="E77">A46</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/>
<a NAME="E63:41"/>
<font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1
 
<b>by </b><i><a class="txt" href="topreal8.html#E41:41"><i><font color="Green" title="E68">A37</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/>
<a NAME="E64:41"/><b>then </b><i><font color="Green" title="E81">A50</font></i>: 
<font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="topreal8.html#E62:41"><i><font color="Green" title="E80">A49</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E65:41"/><i><font color="Green" title="E82">A51</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,<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="c12">g2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E36:41"><i><font color="Green" title="E63">A32</font></i></a>, <a class="txt" href="topreal8.html#E58:41"><i><font color="Green" title="E78">A47</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/>
<a NAME="E66:41"/>
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="c12">g2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E58:41"><i><font color="Green" title="E78">A47</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E67:41"/><b>then </b>
<font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,<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="c12">g2</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E53:41"><i><font color="Green" title="E76">A45</font></i></a>, <a class="ref" href="topreal1.html#T27" title="TOPREAL1:th.27">TOPREAL1:27</a></i>;<br/>
<a NAME="E68:41"/><b>then </b>
not <font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1
 
<b>by </b><i><a class="txt" href="topreal8.html#E65:41"><i><font color="Green" title="E82">A51</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E69:41"/><b>then </b><i><font color="Green" title="E83">A52</font></i>: 
not <font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E70:41"/><i><font color="Green" title="E84">A53</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<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="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c10">g1</font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/>
<a NAME="E71:41"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c12">g2</font>
 
<b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/>
<a NAME="E72:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E60:41"><i><font color="Green" title="E79">A48</font></i></a>, <a class="txt" href="topreal8.html#E70:41"><i><font color="Green" title="E84">A53</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E73:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c10">g1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c12">g2</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E27:41"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="topreal8.html#E39:41"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="topreal8.html#E50:41"><i><font color="Green" title="E75">A44</font></i></a>, <a class="txt" href="topreal8.html#E64:41"><i><font color="Green" title="E81">A50</font></i></a>, <a class="txt" href="topreal8.html#E69:41"><i><font color="Green" title="E83">A52</font></i></a>, <a class="ref" href="topreal8.html#T1" target="_self" title="TOPREAL8:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E74:41"/>
<font color="Maroon" title="c13">g</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green" title="E63">A32</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="topreal8.html#T34" target="_self" title="TOPREAL8:th.34">Th34</a></i>;<br/>

<b>thus </b><a NAME="E75:41"/>
<font color="Maroon" title="c13">g</font> is <a href="goboard5.html#V1" title="GOBOARD5:attr.1">s.c.c.</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="topreal8.html#E27:41"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green" title="E63">A32</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E39:41"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="txt" href="topreal8.html#E50:41"><i><font color="Green" title="E75">A44</font></i></a>, <a class="txt" href="topreal8.html#E60:41"><i><font color="Green" title="E79">A48</font></i></a>, <a class="ref" href="topreal8.html#T33" target="_self" title="TOPREAL8:th.33">Th33</a></i>;<br/>

<b>thus </b><a NAME="E76:41"/>
<font color="Maroon" title="c13">g</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E25:41"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E36:41"><i><font color="Green" title="E63">A32</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="topreal8.html#T26" target="_self" title="TOPREAL8:th.26">Th26</a></i>;<br/>

<b>thus </b> <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f</font> = 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c8">h2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E18:41"><i><font color="Green" title="E50">A13</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="topreal8.html#T35" target="_self" title="TOPREAL8:th.35">Th35</a></i>
<br/>.= 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c13">g</font>
<b>by </b><i><a class="txt" href="topreal8.html#E26:41"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="topreal8.html#E28:41"><i><font color="Green" title="E58">A24</font></i></a>, <a class="txt" href="topreal8.html#E37:41"><i><font color="Green" title="E64">A33</font></i></a>, <a class="txt" href="topreal8.html#E38:41"><i><font color="Green" title="E65">A34</font></i></a>, <a class="txt" href="topreal8.html#E47:41"><i><font color="Green" title="E72">A41</font></i></a>, <a class="ref" href="topreal8.html#T35" target="_self" title="TOPREAL8:th.35">Th35</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c7">h1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="topreal8.html#E48:41"><i><font color="Green" title="E73">A42</font></i></a>, <a class="ref" href="finseq_4.html#T85" title="FINSEQ_4:th.85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
<b>by </b><i><a class="txt" href="topreal8.html#E27:41"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="graph_2.html#T57" title="GRAPH_2:th.57">GRAPH_2:57</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c2">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> = 
<font color="Maroon" title="c8">h2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E13:41"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="topreal8.html#E44:41"><i><font color="Green" title="E70">A39</font></i></a>, <a class="ref" href="finseq_5.html#T30" title="FINSEQ_5:th.30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal8.html#E39:41"><i><font color="Green" title="E66">A35</font></i></a>, <a class="ref" href="graph_2.html#T58" title="GRAPH_2:th.58">GRAPH_2:58</a></i>
;<br/>

<a NAME="E80:41"/><i><font color="Green" title="E85">A54</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E18:41"><i><font color="Green" title="E50">A13</font></i></a>, <a class="ref" href="graph_2.html#T13" title="GRAPH_2:th.13">GRAPH_2:13</a></i>;<br/>
<a NAME="E81:41"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.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="c12">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#T13" title="GRAPH_2:th.13">GRAPH_2:13</a></i>;<br/>
<a NAME="E82:41"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c13">g</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="topreal8.html#E29:41"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="topreal8.html#E40:41"><i><font color="Green" title="E67">A36</font></i></a>, <a class="txt" href="topreal8.html#E80:41"><i><font color="Green" title="E85">A54</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E83:41"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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="c13">g</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>


</div>
