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

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">P1</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">q1</font> &amp; <font color="Maroon">P2</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">p2</font>,<font color="Maroon">q2</font> )
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> holds <br/>( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E3:6"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> holds <br/>( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f1</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<a NAME="E6:6"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E9:6"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">f2</font> <a href="topreal1.html#NR2" target="_self">is_S-Seq</a>  &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="topreal4.html#D1">TOPREAL4:def 1</a></i>;<br/>
<a NAME="E10:6"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E12:6"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f1</font> is <a href="topreal1.html#V1">special</a> &amp; <font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f2</font> is <a href="topreal1.html#V1">special</a> )
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E13:6"/><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 
<b>by </b><i>, , <a class="ref" href="topreal1.html#D10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E14:6"/><b>then </b>
( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:6"/><b>then </b><i><font color="Green">E51</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon">x1</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">y1</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">x2</font> =  <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font>;<br/>
<b>set </b><font color="Maroon">y2</font> =  <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font>;<br/>
<a NAME="E16:6"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E17:6"/><b>then </b><i><font color="Green">E54</font></i>: 
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i>, , , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<a NAME="E18:6"/><i><font color="Green">E55</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard4.html#D1" target="_self">GOBOARD4:def 1</a><br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E3:6_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_1"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:6"/><i><font color="Green">E57</font></i>: 
 <a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard4.html#D1" target="_self">GOBOARD4:def 1</a><br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<a NAME="E2:6_2"/><b>then </b>
( <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E3:6_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_2"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:6"/><i><font color="Green">E59</font></i>: 
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard4.html#D1" target="_self">GOBOARD4:def 1</a><br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:6_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span>
 ;<br/>

<a NAME="E2:6_3"/><b>then </b>
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E3:6_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_3"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:6"/>
 <a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font> <a href="goboard4.html#R1" target="_self">lies_between</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1,<span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard4.html#D1" target="_self">GOBOARD4:def 1</a><br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:6_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<a NAME="E2:6_4"/><b>then </b>
( <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E3:6_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_4"/>
( <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <font color="Maroon">f2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:6"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , , , , , </i>;<br/>


</div>
