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

<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.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/>

<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E36">A1</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f</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="c2">g</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="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c1">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="c2">g</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">g</font></span>)</span>
 <b>and </b><br/><a NAME="E3:38"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c2">g</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="c1">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="c1">f</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon" title="c4">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E4:38"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>and </b><br/><a NAME="E5:38"/><i><font color="Green" title="E40">A5</font></i>: 
( ( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> ) or <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> )
 ;<br/>

<a NAME="E6:38"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:38"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E8:38"/><b>then </b><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal8.html#E5:38"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:38"/><i><font color="Green" title="E43">A8</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="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="E10:38"/><i><font color="Green" title="E44">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="c2">g</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</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="E11:38"/><i><font color="Green" title="E45">A10</font></i>: 
for <font color="Olive" title="b1">i</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 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</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="c1">f</font> holds <br/><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1"/><i><font color="Green" title="E46">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E2:38_1"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c5">i</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="c1">f</font>
 ;<br/>

<a NAME="E3:38_1"/><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c5">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="c1">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c1">f</font> )
 
<b>by </b><i><a class="txt" href="topreal8.html#E1:38_1"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="topreal8.html#E2:38_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E4:38_1"/><b>then </b>
( <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c1">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="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_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="c1">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1"/>
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E2:38_1"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="topreal8.html#E3:38_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_2"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> or <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c1">f</font> or ( <font color="Maroon" title="c4">j</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> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">i</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="c1">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> ) )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_1"><b>suppose </b></a><a NAME="E1:38_2_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:38_2_1"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:38_2_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_2"><b>suppose </b></a><a NAME="E1:38_2_2"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 ;<br/><div class="add"><a NAME="E2:38_2_2"/><b>then </b>
<font color="Maroon" title="c3">i</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="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38"><i><font color="Green" title="E41">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="E3:38_2_2"/><b>then </b><i><font color="Green" title="E47">A15</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="topreal8.html#T28" target="_self" title="TOPREAL8:th.28">Th28</a></i>;<br/><a NAME="E4:38_2_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_2"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="topreal8.html#T28" target="_self" title="TOPREAL8:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E5:38_2_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_2"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_3"><b>suppose </b></a><a NAME="E1:38_2_3"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c3">i</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> <font color="Maroon" title="c1">f</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:38_2_3"/><i><font color="Green" title="E47">A17</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E4:38_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_3"><i><font color="Green" title="E46">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:38_2_3"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c5">m</font>, <font color="Maroon" title="c8">n</font> = <font color="Maroon" title="c6">n</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/><a NAME="E8:38_2_3"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_3"><i><font color="Green" title="E48">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:38_2_3"/>
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="topreal8.html#E9:38"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:38_2_3"/><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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38_2_3"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/><a NAME="E11:38_2_3"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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/><a NAME="E12:38_2_3"/>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38_2_3"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:38_2_3"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E11:38_2_3"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:38_2_3"/><b>then </b><i><font color="Green" title="E51">A21</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="topreal8.html#T31" target="_self" title="TOPREAL8:th.31">Th31</a></i>;<br/><a NAME="E15:38_2_3"/><i><font color="Green" title="E52">A22</font></i>: 
<font color="Maroon" title="c7">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="c8">n</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38_2_3"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:38_2_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E17:38_2_3"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E15:38_2_3"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:38_2_3"/><b>then </b><i><font color="Green" title="E53">A23</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38_2_3"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="topreal8.html#E11:38_2_3"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="topreal8.html#T29" target="_self" title="TOPREAL8:th.29">Th29</a></i>;<br/><a NAME="E19:38_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:38_2_3"/><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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38_2_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_3"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/><a NAME="E21:38_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E22:38_2_3"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E14:38_2_3"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="topreal8.html#E18:38_2_3"><i><font color="Green" title="E53">A23</font></i></a>, <a class="ref" href="topreal1.html#D9" title="TOPREAL1:def.9">TOPREAL1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:38_2_4"/><i><font color="Green" title="E46">A24</font></i>: 
<font color="Maroon" title="c4">j</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> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E2:38_2_4"/><i><font color="Green" title="E47">A25</font></i>: 
<font color="Maroon" title="c3">i</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="c1">f</font>
 <b>and </b><br/><a NAME="E3:38_2_4"/><i><font color="Green" title="E48">A26</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E4:38_2_4"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:38_2_4"/><i><font color="Green" title="E49">A27</font></i>: 
<font color="Maroon" title="c5">x</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></i>;<br/><a NAME="E7:38_2_4"/><i><font color="Green" title="E50">A28</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38_2_4"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="topreal8.html#T28" target="_self" title="TOPREAL8:th.28">Th28</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:38_2_4"/><i><font color="Green" title="E51">A29</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</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/><a NAME="E11:38_2_4"/>
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="topreal8.html#E9:38"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:38_2_4"/><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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E9:38_2_4"><i><font color="Green" title="E51">A29</font></i></a></i>;<br/><a NAME="E13:38_2_4"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">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/><a NAME="E14:38_2_4"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E9:38_2_4"><i><font color="Green" title="E51">A29</font></i></a>, <a class="ref" href="topreal8.html#T31" target="_self" title="TOPREAL8:th.31">Th31</a></i>;<br/><a NAME="E15:38_2_4"/><b>then </b><i><font color="Green" title="E52">A30</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font> <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="c2">g</font>
 <b>by </b><i><a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E16:38_2_4"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <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="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E7:38_2_4"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="topreal3.html#T26" title="TOPREAL3:th.26">TOPREAL3:26</a></i>;<br/><a NAME="E17:38_2_4"/><b>then </b><i><font color="Green" title="E53">A31</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font></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="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#E1:38"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="topreal8.html#E15:38_2_4"><i><font color="Green" title="E52">A30</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_2_4_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 ) or ( <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 ) or ( <font color="Maroon" title="c4">j</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="c1">f</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 ) or ( <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> ) or ( <font color="Maroon" title="c4">j</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="c1">f</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="topreal8.html#E5:38"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4"><i><font color="Green" title="E46">A24</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E17:38_2_4"><i><font color="Green" title="E53">A31</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:38_2_4_1_1_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38_2_4"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E34"><i><font color="Green" title="E33">Lm2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:38_2_4_1_1_2"/><i><font color="Green" title="E54">A32</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:38_2_4_1_1_2"/><i><font color="Green" title="E55">A33</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 ;<br/><div class="add"><a NAME="E3:38_2_4_1_1_2"/><i><font color="Green" title="E56">A34</font></i>: 
 <a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E11:38"><i><font color="Green" title="E45">A10</font></i></a>, <a class="ref" href="topreal8.html#T18" target="_self" title="TOPREAL8:th.18">Th18</a></i>;<br/><a NAME="E4:38_2_4_1_1_2"/><i><font color="Green" title="E57">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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="c1">f</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="E5:38_2_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_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="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">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="c1">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E6:38_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E58">A36</font></i>: 
<font color="Maroon" title="c2">g</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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E4:38_2_4_1_1_2"><i><font color="Green" title="E57">A35</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then </b><i><font color="Green" title="E59">A37</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <font color="Maroon" title="c1">f</font>
<b>by </b><i><a class="ref" href="topreal8.html#T8" target="_self" title="TOPREAL8:th.8">Th8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
<b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="topreal8.html#T6" target="_self" title="TOPREAL8:th.6">Th6</a></i>
;<br/><a NAME="E8:38_2_4_1_1_2"/><i><font color="Green" title="E60">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal8.html#T10" target="_self" title="TOPREAL8:th.10">Th10</a></i>;<br/><a NAME="E9:38_2_4_1_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font> is <a href="finseq_6.html#V1" title="FINSEQ_6:attr.1">circular</a>
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="topreal8.html#T11" target="_self" title="TOPREAL8:th.11">Th11</a></i>;<br/><a NAME="E10:38_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E61">A39</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38_2_4"><i><font color="Green" title="E47">A25</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_4"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4_1_1_2"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="topreal8.html#E7:38_2_4_1_1_2"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="topreal8.html#E8:38_2_4_1_1_2"><i><font color="Green" title="E60">A38</font></i></a>, <a class="ref" href="revrot_1.html#T32" title="REVROT_1:th.32">REVROT_1:32</a></i>;<br/><a NAME="E11:38_2_4_1_1_2"/><i><font color="Green" title="E62">A40</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K1" title="FINSEQ_6:func.1">Rotate</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_2"><i><font color="Green" title="E54">A32</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4_1_1_2"><i><font color="Green" title="E58">A36</font></i></a>, <a class="txt" href="topreal8.html#E7:38_2_4_1_1_2"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="topreal8.html#E8:38_2_4_1_1_2"><i><font color="Green" title="E60">A38</font></i></a>, <a class="ref" href="revrot_1.html#T25" title="REVROT_1:th.25">REVROT_1:25</a></i>;<br/><a NAME="E12:38_2_4_1_1_2"/>
<font color="Maroon" title="c4">j</font> <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:38_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E63">A41</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_2"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:38_2_4_1_1_2"/>
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c2">g</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="topreal8.html#E8:38"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="topreal8.html#E9:38"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="topreal8.html#E10:38"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:38_2_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <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="c2">g</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E10:38"><i><font color="Green" title="E44">A9</font></i></a>, <a class="txt" href="topreal8.html#E13:38_2_4_1_1_2"><i><font color="Green" title="E63">A41</font></i></a>, <a class="ref" href="binarith.html#T72" title="BINARITH:th.72">BINARITH:72</a></i>;<br/><a NAME="E16:38_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E64">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_2"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="binarith.html#T56" title="BINARITH:th.56">BINARITH:56</a></i>;<br/><a NAME="E17:38_2_4_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="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_2"><i><font color="Green" title="E54">A32</font></i></a>, <a class="ref" href="binarith.html#T70" title="BINARITH:th.70">BINARITH:70</a></i>;<br/><a NAME="E18:38_2_4_1_1_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E19:38_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E2:38_2_4_1_1_2"><i><font color="Green" title="E55">A33</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_4_1_1_2"><i><font color="Green" title="E56">A34</font></i></a>, <a class="txt" href="topreal8.html#E7:38_2_4_1_1_2"><i><font color="Green" title="E59">A37</font></i></a>, <a class="txt" href="topreal8.html#E10:38_2_4_1_1_2"><i><font color="Green" title="E61">A39</font></i></a>, <a class="txt" href="topreal8.html#E11:38_2_4_1_1_2"><i><font color="Green" title="E62">A40</font></i></a>, <a class="txt" href="topreal8.html#E16:38_2_4_1_1_2"><i><font color="Green" title="E64">A42</font></i></a>, <a class="txt" href="topreal8.html#E34"><i><font color="Green" title="E33">Lm2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:38_2_4_1_1_3"/><i><font color="Green" title="E54">A43</font></i>: 
<font color="Maroon" title="c4">j</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="c1">f</font>
 <b>and </b><br/><a NAME="E2:38_2_4_1_1_3"/><i><font color="Green" title="E55">A44</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 ;<br/><div class="add"><a NAME="E3:38_2_4_1_1_3"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:38_2_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c3">i</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="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_3"><i><font color="Green" title="E54">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:38_2_4_1_1_3"/><b>then </b><i><font color="Green" title="E56">A45</font></i>: 
<font color="Maroon" title="c3">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="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38_2_4"><i><font color="Green" title="E48">A26</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:38_2_4_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:38_2_4_1_1_3"/><b>then </b><i><font color="Green" title="E57">A46</font></i>: 
<font color="Maroon" title="c3">i</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="c1">f</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_3"><i><font color="Green" title="E54">A43</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:38_2_4_1_1_3"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:38_2_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:38_2_4_1_1_3"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_3"><i><font color="Green" title="E54">A43</font></i></a>, <a class="ref" href="topreal8.html#T28" target="_self" title="TOPREAL8:th.28">Th28</a></i>;<br/><a NAME="E11:38_2_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">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="c1">f</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="c1">f</font>,<font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E2:38_2_4_1_1_3"><i><font color="Green" title="E55">A44</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E12:38_2_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E4:38"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_3"><i><font color="Green" title="E54">A43</font></i></a>, <a class="txt" href="topreal8.html#E5:38_2_4_1_1_3"><i><font color="Green" title="E56">A45</font></i></a>, <a class="txt" href="topreal8.html#E7:38_2_4_1_1_3"><i><font color="Green" title="E57">A46</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:38_2_4_1_1_4"/><i><font color="Green" title="E54">A47</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 <b>and </b><br/><a NAME="E2:38_2_4_1_1_4"/><i><font color="Green" title="E55">A48</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>and </b><br/><a NAME="E3:38_2_4_1_1_4"/><i><font color="Green" title="E56">A49</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 ;<br/><div class="add"><b>thus </b><a NAME="E4:38_2_4_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_4"><i><font color="Green" title="E54">A47</font></i></a>, <a class="txt" href="topreal8.html#E2:38_2_4_1_1_4"><i><font color="Green" title="E55">A48</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_4_1_1_4"><i><font color="Green" title="E56">A49</font></i></a>, <a class="txt" href="topreal8.html#E35"><i><font color="Green" title="E34">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_2_4_1_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:38_2_4_1_1_5"/><i><font color="Green" title="E54">A50</font></i>: 
<font color="Maroon" title="c4">j</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="c1">f</font>
 <b>and </b><br/><a NAME="E2:38_2_4_1_1_5"/><i><font color="Green" title="E55">A51</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <b>and </b><br/><a NAME="E3:38_2_4_1_1_5"/><i><font color="Green" title="E56">A52</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>
 ;<br/><div class="add"><a NAME="E4:38_2_4_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E9:38"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="topreal8.html#E3:38_2_4_1_1_5"><i><font color="Green" title="E56">A52</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:38_2_4_1_1_5"/><b>then </b><i><font color="Green" title="E57">A53</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font></span>)</span>
 ;<br/><a NAME="E6:38_2_4_1_1_5"/><b>then </b><i><font color="Green" title="E58">A54</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_5"><i><font color="Green" title="E54">A50</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:38_2_4_1_1_5"/>
<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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="ref" href="topreal8.html#T2" target="_self" title="TOPREAL8:th.2">Th2</a></i>;<br/><a NAME="E8:38_2_4_1_1_5"/><b>then </b>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c2">g</font></span>)</span>,<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="c1">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">g</font>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="topreal8.html#E3:38"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="topreal8.html#T31" target="_self" title="TOPREAL8:th.31">Th31</a></i>;<br/><a NAME="E9:38_2_4_1_1_5"/><b>then </b><i><font color="Green" title="E59">A55</font></i>: 
<font color="Maroon" title="c2">g</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">g</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="c2">g</font>,1
 <b>by </b><i><a class="txt" href="topreal8.html#E2:38"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="topreal8.html#E6:38_2_4"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="topreal8.html#E1:38_2_4_1_1_5"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topreal8.html#E2:38_2_4_1_1_5"><i><font color="Green" title="E55">A51</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="E10:38_2_4_1_1_5"/><i><font color="Green" title="E60">A56</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="c2">g</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="E11:38_2_4_1_1_5"/>
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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="topreal8.html#E6:38_2_4_1_1_5"><i><font color="Green" title="E58">A54</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E12:38_2_4_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal8.html#E1:38_2_4_1_1_5"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="topreal8.html#E5:38_2_4_1_1_5"><i><font color="Green" title="E57">A53</font></i></a>, <a class="txt" href="topreal8.html#E9:38_2_4_1_1_5"><i><font color="Green" title="E59">A55</font></i></a>, <a class="txt" href="topreal8.html#E10:38_2_4_1_1_5"><i><font color="Green" title="E60">A56</font></i></a>, <a class="ref" href="goboard2.html#T7" title="GOBOARD2:th.7">GOBOARD2:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:38_2_4"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
