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

<b>let </b><font color="Maroon" title="c1">f1</font>, <font color="Maroon" title="c2">f2</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">f1</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp; <font color="Maroon" title="c2">f2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> &amp; 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">f1</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">f1</font></span>)</span> &amp; <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f2</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="c2">f2</font></span>)</span> &amp;  <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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">f1</font></span>)</span> &amp;  <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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">f1</font></span>)</span> &amp;  <a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c2">f2</font></span>)</span> &amp;  <a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c2">f2</font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">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="E3:31"/><i><font color="Green" title="E26">A2</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c3">g1</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp; <font color="Maroon" title="c3">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</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">f1</font> &amp; <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c1">f1</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">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</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="c3">g1</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</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> <font color="Maroon" title="c1">f1</font> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#T29" target="_self" title="JGRAPH_1:th.29">Th29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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="E5:31"/><i><font color="Green" title="E27">A3</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> &amp; <font color="Maroon" title="c4">g2</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> &amp; <font color="Maroon" title="c4">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">g2</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">f2</font> &amp; <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c2">f2</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="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</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="c4">g2</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">g2</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> <font color="Maroon" title="c2">f2</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="jgraph_1.html#T29" target="_self" title="JGRAPH_1:th.29">Th29</a></i>;<br/>
<a NAME="E6:31"/>
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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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><a NAME="E1:31_1"/><i><font color="Green" title="E28">A4</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 ;<br/>

<a NAME="E2:31_1"/><i><font color="Green" title="E29">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E3:31_1"/><i><font color="Green" title="E30">A6</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E4:31_1"/><i><font color="Green" title="E31">A7</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:31_1"/><b>then </b><i><font color="Green" title="E32">A8</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:31_1"/><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c3">g1</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="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:31_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E7:31_1"/><i><font color="Green" title="E34">A10</font></i>: 
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="c1">f1</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:31_1"/><b>then </b><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c3">g1</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="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E6:31_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E9:31_1"/>
<font color="Maroon" title="c3">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="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</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="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:31_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:31_1"/><b>then </b><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c3">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="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</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="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E7:31_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:31_1"/><i><font color="Green" title="E37">A13</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="jgraph_1.html#E7:31_1"><i><font color="Green" title="E34">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="E12:31_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="jgraph_1.html#E3:31_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:31_1"/><b>then </b><i><font color="Green" title="E38">A14</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="c3">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:31_1"/><b>then </b>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <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="c3">g1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E16:31_1"/><i><font color="Green" title="E39">A15</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">y</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="txt" href="jgraph_1.html#E16:31_1"><i><font color="Green" title="E39">A15</font></i></a></i>;<br/>
<a NAME="E18:31_1"/>
<font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E16:31_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E19:31_1"/><b>then </b><i><font color="Green" title="E40">A16</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E13:31_1"><i><font color="Green" title="E38">A14</font></i></a>, <a class="txt" href="jgraph_1.html#E16:31_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E20:31_1"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E16:31_1"><i><font color="Green" title="E39">A15</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:31_1"/><b>then </b><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:31_1"/><b>then </b><i><font color="Green" title="E42">A18</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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">f1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E23:31_1"/><i><font color="Green" title="E43">A19</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E21:31_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E24:31_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E7:31_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:31_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:31_1"/><b>then </b><i><font color="Green" title="E44">A20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E8:31_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="jgraph_1.html#E10:31_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="txt" href="jgraph_1.html#E11:31_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="jgraph_1.html#E19:31_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="txt" href="jgraph_1.html#E22:31_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="txt" href="jgraph_1.html#E23:31_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E27:31_1"/><i><font color="Green" title="E45">A21</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E28:31_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:31_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:31_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E30:31_1"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:31_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="jgraph_1.html#E26:31_1"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="jgraph_1.html#E27:31_1"><i><font color="Green" title="E45">A21</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b><i><font color="Green" title="E28">A22</font></i>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E8:31"/>
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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><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><a NAME="E1:31_2"/><i><font color="Green" title="E29">A23</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 ;<br/>

<a NAME="E2:31_2"/><i><font color="Green" title="E30">A24</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E3:31_2"/><i><font color="Green" title="E31">A25</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E4:31_2"/><i><font color="Green" title="E32">A26</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E5:31_2"/><i><font color="Green" title="E33">A27</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E6:31_2"/><i><font color="Green" title="E34">A28</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:31_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:31_2"/><b>then </b><i><font color="Green" title="E35">A29</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:31_2"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c3">g1</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="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:31_2"/><i><font color="Green" title="E37">A31</font></i>: 
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="c1">f1</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:31_2"/><b>then </b><i><font color="Green" title="E38">A32</font></i>: 
<font color="Maroon" title="c3">g1</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="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E9:31_2"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:31_2"/>
<font color="Maroon" title="c3">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="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</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="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E13:31_2"/><b>then </b><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c3">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="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</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="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E10:31_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E14:31_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E10:31_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:31_2"/><b>then </b><i><font color="Green" title="E40">A34</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> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_2"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:31_2"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="txt" href="jgraph_1.html#E4:31_2"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:31_2"/><b>then </b><i><font color="Green" title="E41">A35</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="c4">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:31_2"/><b>then </b>
<font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <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="c4">g2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:31_2"/><i><font color="Green" title="E42">A36</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">y</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="txt" href="jgraph_1.html#E20:31_2"><i><font color="Green" title="E42">A36</font></i></a></i>;<br/>
<a NAME="E22:31_2"/>
<font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E20:31_2"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E23:31_2"/><b>then </b><i><font color="Green" title="E43">A37</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E17:31_2"><i><font color="Green" title="E41">A35</font></i></a>, <a class="txt" href="jgraph_1.html#E20:31_2"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E24:31_2"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E20:31_2"><i><font color="Green" title="E42">A36</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:31_2"/><b>then </b><i><font color="Green" title="E44">A38</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_2"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:31_2"/><b>then </b><i><font color="Green" title="E45">A39</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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">f1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E27:31_2"/><i><font color="Green" title="E46">A40</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E25:31_2"><i><font color="Green" title="E44">A38</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E28:31_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E10:31_2"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:31_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_2"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:31_2"/><b>then </b><i><font color="Green" title="E47">A41</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:31_2"><i><font color="Green" title="E38">A32</font></i></a>, <a class="txt" href="jgraph_1.html#E13:31_2"><i><font color="Green" title="E39">A33</font></i></a>, <a class="txt" href="jgraph_1.html#E15:31_2"><i><font color="Green" title="E40">A34</font></i></a>, <a class="txt" href="jgraph_1.html#E23:31_2"><i><font color="Green" title="E43">A37</font></i></a>, <a class="txt" href="jgraph_1.html#E26:31_2"><i><font color="Green" title="E45">A39</font></i></a>, <a class="txt" href="jgraph_1.html#E27:31_2"><i><font color="Green" title="E46">A40</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E31:31_2"/><i><font color="Green" title="E48">A42</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_2"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E32:31_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_2"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E33:31_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31_2"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E34:31_2"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E8:31_2"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="jgraph_1.html#E30:31_2"><i><font color="Green" title="E47">A41</font></i></a>, <a class="txt" href="jgraph_1.html#E31:31_2"><i><font color="Green" title="E48">A42</font></i></a>, <a class="ref" href="goboard1.html#D3" title="GOBOARD1:def.3">GOBOARD1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:31"/><b>then </b><i><font color="Green" title="E29">A43</font></i>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c4">g2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">X_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E10:31"/>
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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><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><a NAME="E1:31_3"/><i><font color="Green" title="E30">A44</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 ;<br/>

<a NAME="E2:31_3"/><i><font color="Green" title="E31">A45</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E3:31_3"/><i><font color="Green" title="E32">A46</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E4:31_3"/><i><font color="Green" title="E33">A47</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E5:31_3"/><i><font color="Green" title="E34">A48</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E6:31_3"/><i><font color="Green" title="E35">A49</font></i>: 
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="c4">g2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:31_3"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:31_3"/><b>then </b><i><font color="Green" title="E36">A50</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31_3"><i><font color="Green" title="E34">A48</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:31_3"/><i><font color="Green" title="E37">A51</font></i>: 
<font color="Maroon" title="c4">g2</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="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_3"><i><font color="Green" title="E35">A49</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:31_3"/><i><font color="Green" title="E38">A52</font></i>: 
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">f2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:31_3"/><b>then </b><i><font color="Green" title="E39">A53</font></i>: 
<font color="Maroon" title="c4">g2</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">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E9:31_3"><i><font color="Green" title="E37">A51</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:31_3"/>
<font color="Maroon" title="c4">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="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</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="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_3"><i><font color="Green" title="E35">A49</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E13:31_3"/><b>then </b><i><font color="Green" title="E40">A54</font></i>: 
<font color="Maroon" title="c4">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="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</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">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E10:31_3"><i><font color="Green" title="E38">A52</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E14:31_3"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E10:31_3"><i><font color="Green" title="E38">A52</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:31_3"/><b>then </b><i><font color="Green" title="E41">A55</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_3"><i><font color="Green" title="E32">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:31_3"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">g1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_3"><i><font color="Green" title="E30">A44</font></i></a>, <a class="txt" href="jgraph_1.html#E4:31_3"><i><font color="Green" title="E33">A47</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:31_3"/><b>then </b><i><font color="Green" title="E42">A56</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="c3">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:31_3"/><b>then </b>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <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="c3">g1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:31_3"/><i><font color="Green" title="E43">A57</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">y</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="txt" href="jgraph_1.html#E20:31_3"><i><font color="Green" title="E43">A57</font></i></a></i>;<br/>
<a NAME="E22:31_3"/>
<font color="Maroon" title="c1">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E20:31_3"><i><font color="Green" title="E43">A57</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E23:31_3"/><b>then </b><i><font color="Green" title="E44">A58</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f1</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E17:31_3"><i><font color="Green" title="E42">A56</font></i></a>, <a class="txt" href="jgraph_1.html#E20:31_3"><i><font color="Green" title="E43">A57</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E24:31_3"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E20:31_3"><i><font color="Green" title="E43">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:31_3"/><b>then </b><i><font color="Green" title="E45">A59</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_3"><i><font color="Green" title="E31">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:31_3"/><b>then </b><i><font color="Green" title="E46">A60</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c2">f2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E27:31_3"/><i><font color="Green" title="E47">A61</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c1">f1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E25:31_3"><i><font color="Green" title="E45">A59</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E28:31_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E10:31_3"><i><font color="Green" title="E38">A52</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:31_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_3"><i><font color="Green" title="E32">A46</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:31_3"/><b>then </b><i><font color="Green" title="E48">A62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E11:31_3"><i><font color="Green" title="E39">A53</font></i></a>, <a class="txt" href="jgraph_1.html#E13:31_3"><i><font color="Green" title="E40">A54</font></i></a>, <a class="txt" href="jgraph_1.html#E15:31_3"><i><font color="Green" title="E41">A55</font></i></a>, <a class="txt" href="jgraph_1.html#E23:31_3"><i><font color="Green" title="E44">A58</font></i></a>, <a class="txt" href="jgraph_1.html#E26:31_3"><i><font color="Green" title="E46">A60</font></i></a>, <a class="txt" href="jgraph_1.html#E27:31_3"><i><font color="Green" title="E47">A61</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E31:31_3"/><i><font color="Green" title="E49">A63</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_3"><i><font color="Green" title="E30">A44</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E32:31_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_3"><i><font color="Green" title="E35">A49</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E33:31_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31_3"><i><font color="Green" title="E34">A48</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E34:31_3"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E8:31_3"><i><font color="Green" title="E36">A50</font></i></a>, <a class="txt" href="jgraph_1.html#E30:31_3"><i><font color="Green" title="E48">A62</font></i></a>, <a class="txt" href="jgraph_1.html#E31:31_3"><i><font color="Green" title="E49">A63</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:31"/><b>then </b><i><font color="Green" title="E30">A64</font></i>: 
 <a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c3">g1</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E12:31"/>
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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><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><a NAME="E1:31_4"/><i><font color="Green" title="E31">A65</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 ;<br/>

<a NAME="E2:31_4"/><i><font color="Green" title="E32">A66</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E3:31_4"/><i><font color="Green" title="E33">A67</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E4:31_4"/><i><font color="Green" title="E34">A68</font></i>: 
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="c4">g2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:31_4"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:31_4"/><b>then </b><i><font color="Green" title="E35">A69</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_4"><i><font color="Green" title="E33">A67</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:31_4"/><i><font color="Green" title="E36">A70</font></i>: 
<font color="Maroon" title="c4">g2</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="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:31_4"><i><font color="Green" title="E34">A68</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:31_4"/><i><font color="Green" title="E37">A71</font></i>: 
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">f2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:31_4"/><b>then </b><i><font color="Green" title="E38">A72</font></i>: 
<font color="Maroon" title="c4">g2</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">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E7:31_4"><i><font color="Green" title="E36">A70</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:31_4"/>
<font color="Maroon" title="c4">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="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</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="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E4:31_4"><i><font color="Green" title="E34">A68</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E11:31_4"/><b>then </b><i><font color="Green" title="E39">A73</font></i>: 
<font color="Maroon" title="c4">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="c4">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</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">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E8:31_4"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E12:31_4"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E8:31_4"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:31_4"/><b>then </b><i><font color="Green" title="E40">A74</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> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_4"><i><font color="Green" title="E32">A66</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:31_4"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_4"><i><font color="Green" title="E31">A65</font></i></a>, <a class="txt" href="jgraph_1.html#E3:31_4"><i><font color="Green" title="E33">A67</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E15:31_4"/><b>then </b><i><font color="Green" title="E41">A75</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="c4">g2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:31_4"/><b>then </b>
<font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <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="c4">g2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:31_4"/><i><font color="Green" title="E42">A76</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c6">y</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="txt" href="jgraph_1.html#E18:31_4"><i><font color="Green" title="E42">A76</font></i></a></i>;<br/>
<a NAME="E20:31_4"/>
<font color="Maroon" title="c2">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E18:31_4"><i><font color="Green" title="E42">A76</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E21:31_4"/><b>then </b><i><font color="Green" title="E43">A77</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f2</font> &amp; <font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E15:31_4"><i><font color="Green" title="E41">A75</font></i></a>, <a class="txt" href="jgraph_1.html#E18:31_4"><i><font color="Green" title="E42">A76</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E22:31_4"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E18:31_4"><i><font color="Green" title="E42">A76</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E23:31_4"/><b>then </b><i><font color="Green" title="E44">A78</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_4"><i><font color="Green" title="E32">A66</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:31_4"/><b>then </b><i><font color="Green" title="E45">A79</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c2">f2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E25:31_4"/><i><font color="Green" title="E46">A80</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E23:31_4"><i><font color="Green" title="E44">A78</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E26:31_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E8:31_4"><i><font color="Green" title="E37">A71</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E27:31_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c2">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E2:31_4"><i><font color="Green" title="E32">A66</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:31_4"/><b>then </b><i><font color="Green" title="E47">A81</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:31_4"><i><font color="Green" title="E38">A72</font></i></a>, <a class="txt" href="jgraph_1.html#E11:31_4"><i><font color="Green" title="E39">A73</font></i></a>, <a class="txt" href="jgraph_1.html#E13:31_4"><i><font color="Green" title="E40">A74</font></i></a>, <a class="txt" href="jgraph_1.html#E21:31_4"><i><font color="Green" title="E43">A77</font></i></a>, <a class="txt" href="jgraph_1.html#E24:31_4"><i><font color="Green" title="E45">A79</font></i></a>, <a class="txt" href="jgraph_1.html#E25:31_4"><i><font color="Green" title="E46">A80</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E29:31_4"/><i><font color="Green" title="E48">A82</font></i>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_4"><i><font color="Green" title="E31">A65</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E30:31_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31_4"><i><font color="Green" title="E33">A67</font></i></a>, <a class="txt" href="jgraph_1.html#E4:31_4"><i><font color="Green" title="E34">A68</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E31:31_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E32:31_4"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E6:31_4"><i><font color="Green" title="E35">A69</font></i></a>, <a class="txt" href="jgraph_1.html#E28:31_4"><i><font color="Green" title="E47">A81</font></i></a>, <a class="txt" href="jgraph_1.html#E29:31_4"><i><font color="Green" title="E48">A82</font></i></a>, <a class="ref" href="goboard1.html#D4" title="GOBOARD1:def.4">GOBOARD1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:31"/><b>then </b><i><font color="Green" title="E31">A83</font></i>: 
 <a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font> <a href="goboard4.html#R1" title="GOBOARD4:pred.1">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3" title="GOBOARD1:func.3">Y_axis</a> <font color="Maroon" title="c4">g2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c4">g2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="goboard4.html#D2" title="GOBOARD4:def.2">GOBOARD4:def 2</a></i>;<br/>
<a NAME="E14:31"/><i><font color="Green" title="E32">A84</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">k</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="c3">g1</font> holds <br/><font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_5"/><i><font color="Green" title="E33">A85</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c5">k</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="c3">g1</font> )
 ;<br/>

<a NAME="E2:31_5"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:31_5"/><b>then </b><i><font color="Green" title="E34">A86</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E1:31_5"><i><font color="Green" title="E33">A85</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E4:31_5"/>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_5"><i><font color="Green" title="E33">A85</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:31_5"/>
<font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">g1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_5"><i><font color="Green" title="E33">A85</font></i></a>, <a class="txt" href="jgraph_1.html#E3:31_5"><i><font color="Green" title="E34">A86</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:31"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g2</font> &amp; <font color="Olive" title="b1">k</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="c4">g2</font> holds <br/><font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:31_6"/><i><font color="Green" title="E33">A87</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g2</font> &amp; <font color="Maroon" title="c5">k</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="c4">g2</font> )
 ;<br/>

<a NAME="E2:31_6"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:31_6"/><b>then </b><i><font color="Green" title="E34">A88</font></i>: 
<font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E1:31_6"><i><font color="Green" title="E33">A87</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E4:31_6"/>
<font color="Maroon" title="c4">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_6"><i><font color="Green" title="E33">A87</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:31_6"/>
<font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">g2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_1.html#E1:31_6"><i><font color="Green" title="E33">A87</font></i></a>, <a class="txt" href="jgraph_1.html#E3:31_6"><i><font color="Green" title="E34">A88</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:31"/><b>then </b>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">g1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">g2</font>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="jgraph_1.html#E7:31"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="jgraph_1.html#E9:31"><i><font color="Green" title="E29">A43</font></i></a>, <a class="txt" href="jgraph_1.html#E11:31"><i><font color="Green" title="E30">A64</font></i></a>, <a class="txt" href="jgraph_1.html#E13:31"><i><font color="Green" title="E31">A83</font></i></a>, <a class="txt" href="jgraph_1.html#E14:31"><i><font color="Green" title="E32">A84</font></i></a>, <a class="ref" href="goboard4.html#T4" title="GOBOARD4:th.4">GOBOARD4:4</a></i>;<br/>
<a NAME="E17:31"/><b>then </b><i><font color="Green" title="E33">A89</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">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="c4">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E18:31"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c3">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="c4">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f1</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">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:31"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="jgraph_1.html#E5:31"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E19:31"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f1</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">f2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E17:31"><i><font color="Green" title="E33">A89</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:31"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c1">f1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">f2</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>
