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

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>set </b><font color="Maroon" title="c7">K</font> =  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>;<br/>
<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E68">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 ;<br/>

<a NAME="E2:79"/><b>then </b><i><font color="Green" title="E69">A2</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E70">A3</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/>
<b>thus </b><a NAME="E4:79"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> implies ( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E71">A4</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 ;<br/>

<a NAME="E2:79_1"/><b>then </b><i><font color="Green" title="E72">A5</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:79"><i><font color="Green" title="E69">A2</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sppol_2.html#D3" title="SPPOL_2:def.3">SPPOL_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
;<br/>
<a NAME="E4:79_1"/><b>then </b>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_1"><i><font color="Green" title="E72">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:79_1"/><b>then </b><i><font color="Green" title="E73">A6</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_1_2_1"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E5:79_1"><i><font color="Green" title="E73">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2_1_1"><b>case </b></a><a NAME="E1:79_1_2_1_1"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:79_1_2_1_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><b>hence </b><a NAME="E3:79_1_2_1_1"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2_1_2"><b>case </b></a><a NAME="E1:79_1_2_1_2"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:79_1_2_1_2"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self" title="JGRAPH_6:th.70">Th70</a></i>;<br/><b>hence </b><a NAME="E3:79_1_2_1_2"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2_1_3"><b>case </b></a><a NAME="E1:79_1_2_1_3"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:79_1_2_1_3"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T71" target="_self" title="JGRAPH_6:th.71">Th71</a></i>;<br/><b>hence </b><a NAME="E3:79_1_2_1_3"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_1_2_1_4"><b>case </b></a><a NAME="E1:79_1_2_1_4"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:79_1_2_1_4"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_1"><i><font color="Green" title="E71">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T68" target="_self" title="JGRAPH_6:th.68">Th68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:79_1"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:79"/>
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_2"/><i><font color="Green" title="E71">A7</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_2_1_1"/>
( ( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  ) or ( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2"><i><font color="Green" title="E71">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1"><b>case </b></a><a NAME="E1:79_2_1_1_1"/><i><font color="Green" title="E72">A8</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_1"/><b>then </b><i><font color="Green" title="E73">A9</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:79_2_1_1_1"/><i><font color="Green" title="E74">A10</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T62" target="_self" title="JGRAPH_6:th.62">Th62</a></i>;<br/><a NAME="E4:79_2_1_1_1"/><b>then </b><i><font color="Green" title="E75">A11</font></i>: 
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:79_2_1_1_1"/><i><font color="Green" title="E76">A12</font></i>: 
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1"><i><font color="Green" title="E74">A10</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:79_2_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1_1_1_1"/>
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font> &amp; <font color="Olive" title="b3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c10">s2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:79_2_1_1_1_1_1"/><i><font color="Green" title="E77">A13</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s1</font> &amp; <font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font> &amp; <font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:79_2_1_1_1_1_1"/><i><font color="Green" title="E78">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:79_2_1_1_1_1_1"/><i><font color="Green" title="E79">A15</font></i>: 
<font color="Maroon" title="c8">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E4:79_2_1_1_1_1_1"/><i><font color="Green" title="E80">A16</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">g1</font> = <font color="Maroon" title="c8">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<a NAME="E6:79_2_1_1_1_1_1"/>
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:79_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E81">A17</font></i>: 
<font color="Maroon" title="c11">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">h1</font> =  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">h2</font> =  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:79_2_1_1_1_1_1"/><i><font color="Green" title="E82">A18</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>;<br/>
<a NAME="E11:79_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E83">A19</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon" title="c12">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> ) implies <font color="Maroon" title="c12">h1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T39" title="JGRAPH_2:th.39">JGRAPH_2:39</a></i>;<br/>
<a NAME="E12:79_2_1_1_1_1_1"/>
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon" title="c13">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> ) implies <font color="Maroon" title="c13">h2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E10:79_2_1_1_1_1_1"><i><font color="Green" title="E82">A18</font></i></a>, <a class="ref" href="jgraph_2.html#T40" title="JGRAPH_2:th.40">JGRAPH_2:40</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E14:79_2_1_1_1_1_1"/><i><font color="Green" title="E84">A20</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c14">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:79_2_1_1_1_1_1"><i><font color="Green" title="E83">A19</font></i></a>, <a class="ref" href="jgraph_2.html#T29" title="JGRAPH_2:th.29">JGRAPH_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">g1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E16:79_2_1_1_1_1_1"/><i><font color="Green" title="E85">A21</font></i>: 
<font color="Maroon" title="c15">k</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_1_1_1"><i><font color="Green" title="E81">A17</font></i></a>, <a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:79_2_1_1_1_1_1"/><i><font color="Green" title="E86">A22</font></i>: 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E87">A23</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font>
 ;<br/><a NAME="E2:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E88">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:79_2_1_1_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green" title="E89">A25</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1"><i><font color="Green" title="E88">A24</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E17:79_2_1_1_1_1_1"><i><font color="Green" title="E86">A22</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E17:79_2_1_1_1_1_1"><i><font color="Green" title="E86">A22</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
;<br/><a NAME="E5:79_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green" title="E90">A26</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1"><i><font color="Green" title="E88">A24</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E3:79"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E7:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E91">A27</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1"><i><font color="Green" title="E88">A24</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1"><i><font color="Green" title="E73">A9</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E9:79_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E92">A28</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1"><i><font color="Green" title="E89">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E6:79_2_1_1_1_1_1_1"><i><font color="Green" title="E90">A26</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E10:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E93">A29</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E11:79_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E94">A30</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">s2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_1_1_1_1"><i><font color="Green" title="E91">A27</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">s2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_1_1_1_1"><i><font color="Green" title="E91">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E10:79_2_1_1_1_1_1_1"><i><font color="Green" title="E93">A29</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><a NAME="E13:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E95">A31</font></i>: 
<font color="Maroon" title="c16">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="txt" href="jgraph_6.html#E7:79_2_1_1_1_1_1_1"><i><font color="Green" title="E91">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E10:79_2_1_1_1_1_1_1"><i><font color="Green" title="E93">A29</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="borsuk_4.html#T49" title="BORSUK_4:th.49">BORSUK_4:49</a></i>;<br/><a NAME="E14:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E96">A32</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E15:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E97">A33</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E98">A34</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E17:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E99">A35</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E18:79_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">xc</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E20:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E100">A36</font></i>: 
( <font color="Maroon" title="c17">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E16:79_2_1_1_1_1_1"><i><font color="Green" title="E85">A21</font></i></a>, <a class="txt" href="jgraph_6.html#E9:79_2_1_1_1_1_1_1"><i><font color="Green" title="E92">A28</font></i></a>, <a class="txt" href="jgraph_6.html#E13:79_2_1_1_1_1_1_1"><i><font color="Green" title="E95">A31</font></i></a>, <a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1_1"><i><font color="Green" title="E96">A32</font></i></a>, <a class="txt" href="jgraph_6.html#E15:79_2_1_1_1_1_1_1"><i><font color="Green" title="E97">A33</font></i></a>, <a class="txt" href="jgraph_6.html#E16:79_2_1_1_1_1_1_1"><i><font color="Green" title="E98">A34</font></i></a>, <a class="txt" href="jgraph_6.html#E17:79_2_1_1_1_1_1_1"><i><font color="Green" title="E99">A35</font></i></a>, <a class="ref" href="topreal5.html#T11" title="TOPREAL5:th.11">TOPREAL5:11</a></i>;<br/><a NAME="E21:79_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">rxc</font> = <font color="Maroon" title="c17">xc</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E23:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E101">A37</font></i>: 
<font color="Maroon" title="c15">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1_1_1_1_1_1_4"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1_4_1"/><i><font color="Green" title="E102">A38</font></i>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c21">r1</font> = <font color="Maroon" title="c19">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">r2</font> = <font color="Maroon" title="c20">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a></i>;<br/>
<i><font color="Green" title="E103">A39</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/>
<i><font color="Green" title="E104">A40</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_1_1_1"><i><font color="Green" title="E84">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E6:79_2_1_1_1_1_1_1_4_1"/><i><font color="Green" title="E105">A41</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1"><i><font color="Green" title="E80">A16</font></i></a></i>;<br/>
<a NAME="E7:79_2_1_1_1_1_1_1_4_1"/><i><font color="Green" title="E106">A42</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1"><i><font color="Green" title="E80">A16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">gr1</font> = <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jgraph_6.html#E6:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E105">A41</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">gr2</font> = <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E106">A42</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1"/>
( ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_1"><i><font color="Green" title="E74">A10</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1"><i><font color="Green" title="E80">A16</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_1"><b>case </b></a><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_1"/><i><font color="Green" title="E107">A43</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E108">A44</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_1"/>
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_1"><i><font color="Green" title="E107">A43</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E4:79_2_1_1_1_1_1_1_4_1_3_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_1"><i><font color="Green" title="E108">A44</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E5:79_2_1_1_1_1_1_1_4_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E6:79_2_1_1_1_1_1_1_4_1_3_1_1"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1"><i><font color="Green" title="E78">A14</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1"><i><font color="Green" title="E79">A15</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_2"><b>case </b></a><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_2"/><i><font color="Green" title="E107">A45</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E108">A46</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_2"/><i><font color="Green" title="E109">A47</font></i>: 
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E107">A45</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_1_1_1_1_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E110">A48</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E107">A45</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><div><i><font color="Green" title="E111">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_2_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A47</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A46</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A47</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_2_2"/>
<font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 ;<br/><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A46</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A47</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E110">A48</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:79_2_1_1_1_1_1_1_4_1_3_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A46</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A47</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1_3_1_2"><i><font color="Green" title="E111">A49</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:79_2_1_1_1_1_1_1_4_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:79_2_1_1_1_1_1_1_4_1_3_1_2"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1"><i><font color="Green" title="E78">A14</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1"><i><font color="Green" title="E79">A15</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_3"><b>case </b></a><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E107">A50</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E108">A51</font></i>: 
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E109">A52</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E107">A50</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_1_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E110">A53</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E107">A50</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A51</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><div><i><font color="Green" title="E111">A54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_3_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_3_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A51</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E110">A53</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_3_2"/>
<font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 ;<br/><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_3_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A51</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A51</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A52</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:79_2_1_1_1_1_1_1_4_1_3_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A51</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A52</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1_3_1_3"><i><font color="Green" title="E111">A54</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:79_2_1_1_1_1_1_1_4_1_3_1_3"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:79_2_1_1_1_1_1_1_4_1_3_1_3"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1"><i><font color="Green" title="E78">A14</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1"><i><font color="Green" title="E79">A15</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_1_1_1_1_4_1_3_1_4"><b>case </b></a><a NAME="E1:79_2_1_1_1_1_1_1_4_1_3_1_4"/><i><font color="Green" title="E107">A55</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_1_1_1_1_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E108">A56</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:79_2_1_1_1_1_1_1_4_1_3_1_4"/>
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1_3_1_4"><i><font color="Green" title="E107">A55</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_1_1_1_1_4_1_3_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E102">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E103">A39</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1_1_1_1_4_1"><i><font color="Green" title="E104">A40</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1_1_4_1_3_1_4"><i><font color="Green" title="E108">A56</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E5:79_2_1_1_1_1_1_1_4_1_3_1_4"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E6:79_2_1_1_1_1_1_1_4_1_3_1_4"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_1_1_1"><i><font color="Green" title="E78">A14</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_1_1_1"><i><font color="Green" title="E79">A15</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:79_2_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_1_1_1_1_1_1_4"/>
<font color="Maroon" title="c15">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:79_2_1_1_1_1_1_1"/><i><font color="Green" title="E102">A57</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E25:79_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1"><i><font color="Green" title="E77">A13</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E26:79_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">rxc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s1</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:79_2_1_1_1_1_1_1"><i><font color="Green" title="E94">A30</font></i></a>, <a class="txt" href="jgraph_6.html#E20:79_2_1_1_1_1_1_1"><i><font color="Green" title="E100">A36</font></i></a>, <a class="txt" href="jgraph_6.html#E23:79_2_1_1_1_1_1_1"><i><font color="Green" title="E101">A37</font></i></a>, <a class="txt" href="jgraph_6.html#E24:79_2_1_1_1_1_1_1"><i><font color="Green" title="E102">A57</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E27:79_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_1_1_1_1"><i><font color="Green" title="E87">A23</font></i></a>, <a class="txt" href="jgraph_6.html#E20:79_2_1_1_1_1_1_1"><i><font color="Green" title="E100">A36</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:79_2_1_1_1_1_1"/>
<font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_1_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_1"><i><font color="Green" title="E75">A11</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1"><i><font color="Green" title="E76">A12</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:79_2_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_1"><i><font color="Green" title="E75">A11</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_1"><i><font color="Green" title="E76">A12</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2"><b>case </b></a><a NAME="E1:79_2_1_1_2"/><i><font color="Green" title="E72">A58</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c6">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_2"/><b>then </b><i><font color="Green" title="E73">A59</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:79_2_1_1_2"/><i><font color="Green" title="E74">A60</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T62" target="_self" title="JGRAPH_6:th.62">Th62</a></i>;<br/><a NAME="E4:79_2_1_1_2"/><b>then </b><i><font color="Green" title="E75">A61</font></i>: 
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2"><i><font color="Green" title="E72">A58</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:79_2_1_1_2"/><i><font color="Green" title="E76">A62</font></i>: 
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2"><i><font color="Green" title="E74">A60</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:79_2_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1_1_2_1"/>
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s1</font> &amp; <font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font> &amp; <font color="Olive" title="b3">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Olive" title="b2">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">s1</font>, <font color="Maroon" title="c10">s2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:79_2_1_1_2_1_1"/><i><font color="Green" title="E77">A63</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s1</font> &amp; <font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font> &amp; <font color="Maroon" title="c10">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:79_2_1_1_2_1_1"/><i><font color="Green" title="E78">A64</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:79_2_1_1_2_1_1"/><i><font color="Green" title="E79">A65</font></i>: 
<font color="Maroon" title="c8">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E4:79_2_1_1_2_1_1"/><i><font color="Green" title="E80">A66</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">g1</font> = <font color="Maroon" title="c8">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<a NAME="E6:79_2_1_1_2_1_1"/>
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:79_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E81">A67</font></i>: 
<font color="Maroon" title="c11">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">h1</font> =  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">h2</font> =  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:79_2_1_1_2_1_1"/><i><font color="Green" title="E82">A68</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3" title="TSEP_1:th.3">TSEP_1:3</a></i>;<br/>
<a NAME="E11:79_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E83">A69</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon" title="c12">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> ) implies <font color="Maroon" title="c12">h1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T39" title="JGRAPH_2:th.39">JGRAPH_2:39</a></i>;<br/>
<a NAME="E12:79_2_1_1_2_1_1"/>
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span>)</span> holds  <font color="Maroon" title="c13">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> ) implies <font color="Maroon" title="c13">h2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E10:79_2_1_1_2_1_1"><i><font color="Green" title="E82">A68</font></i></a>, <a class="ref" href="jgraph_2.html#T40" title="JGRAPH_2:th.40">JGRAPH_2:40</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E14:79_2_1_1_2_1_1"/><i><font color="Green" title="E84">A70</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c14">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:79_2_1_1_2_1_1"><i><font color="Green" title="E83">A69</font></i></a>, <a class="ref" href="jgraph_2.html#T29" title="JGRAPH_2:th.29">JGRAPH_2:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">g1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/>
<a NAME="E16:79_2_1_1_2_1_1"/><i><font color="Green" title="E85">A71</font></i>: 
<font color="Maroon" title="c15">k</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_2_1_1"><i><font color="Green" title="E81">A67</font></i></a>, <a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:79_2_1_1_2_1_1"/><i><font color="Green" title="E86">A72</font></i>: 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E87">A73</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">s2</font>
 ;<br/><a NAME="E2:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E88">A74</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:79_2_1_1_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green" title="E89">A75</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E17:79_2_1_1_2_1_1"><i><font color="Green" title="E86">A72</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E17:79_2_1_1_2_1_1"><i><font color="Green" title="E86">A72</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
;<br/><a NAME="E5:79_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><i><font color="Green" title="E90">A76</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E3:79"><i><font color="Green" title="E70">A3</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E7:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E91">A77</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>then </b><font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1"><i><font color="Green" title="E88">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2"><i><font color="Green" title="E73">A59</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E9:79_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E92">A78</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2"><i><font color="Green" title="E71">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1"><i><font color="Green" title="E89">A75</font></i></a>, <a class="txt" href="jgraph_6.html#E6:79_2_1_1_2_1_1_1"><i><font color="Green" title="E90">A76</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E10:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E93">A79</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E11:79_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A80</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">s2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A77</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c10">s2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A77</font></i></a>, <a class="txt" href="jgraph_6.html#E10:79_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A79</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><a NAME="E13:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E95">A81</font></i>: 
<font color="Maroon" title="c16">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="txt" href="jgraph_6.html#E7:79_2_1_1_2_1_1_1"><i><font color="Green" title="E91">A77</font></i></a>, <a class="txt" href="jgraph_6.html#E10:79_2_1_1_2_1_1_1"><i><font color="Green" title="E93">A79</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="borsuk_4.html#T49" title="BORSUK_4:th.49">BORSUK_4:49</a></i>;<br/><a NAME="E14:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E96">A82</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E15:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E97">A83</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E98">A84</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E17:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E99">A85</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E18:79_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">xc</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E20:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E100">A86</font></i>: 
( <font color="Maroon" title="c17">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">B</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">xc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E16:79_2_1_1_2_1_1"><i><font color="Green" title="E85">A71</font></i></a>, <a class="txt" href="jgraph_6.html#E9:79_2_1_1_2_1_1_1"><i><font color="Green" title="E92">A78</font></i></a>, <a class="txt" href="jgraph_6.html#E13:79_2_1_1_2_1_1_1"><i><font color="Green" title="E95">A81</font></i></a>, <a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1_1"><i><font color="Green" title="E96">A82</font></i></a>, <a class="txt" href="jgraph_6.html#E15:79_2_1_1_2_1_1_1"><i><font color="Green" title="E97">A83</font></i></a>, <a class="txt" href="jgraph_6.html#E16:79_2_1_1_2_1_1_1"><i><font color="Green" title="E98">A84</font></i></a>, <a class="txt" href="jgraph_6.html#E17:79_2_1_1_2_1_1_1"><i><font color="Green" title="E99">A85</font></i></a>, <a class="ref" href="topreal5.html#T11" title="TOPREAL5:th.11">TOPREAL5:11</a></i>;<br/><a NAME="E21:79_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c17">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">rxc</font> = <font color="Maroon" title="c17">xc</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E23:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E101">A87</font></i>: 
<font color="Maroon" title="c15">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:79_2_1_1_2_1_1_1_4"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1_4_1"/><i><font color="Green" title="E102">A88</font></i>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c21">r1</font> = <font color="Maroon" title="c19">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">r2</font> = <font color="Maroon" title="c20">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a></i>;<br/>
<i><font color="Green" title="E103">A89</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/>
<i><font color="Green" title="E104">A90</font></i>: <font color="Maroon" title="c15">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> = 
<font color="Maroon" title="c14">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">h1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_6.html#E14:79_2_1_1_2_1_1"><i><font color="Green" title="E84">A70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/>
<a NAME="E6:79_2_1_1_2_1_1_1_4_1"/><i><font color="Green" title="E105">A91</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1"><i><font color="Green" title="E80">A66</font></i></a></i>;<br/>
<a NAME="E7:79_2_1_1_2_1_1_1_4_1"/><i><font color="Green" title="E106">A92</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1"><i><font color="Green" title="E80">A66</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">gr1</font> = <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jgraph_6.html#E6:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E105">A91</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">gr2</font> = <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jgraph_6.html#E7:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E106">A92</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1"/>
( ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or ( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_2"><i><font color="Green" title="E74">A60</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1"><i><font color="Green" title="E80">A66</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_1"><b>case </b></a><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_1"/><i><font color="Green" title="E107">A93</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_1"/><b>then </b><i><font color="Green" title="E108">A94</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_1"/>
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_1"><i><font color="Green" title="E107">A93</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E4:79_2_1_1_2_1_1_1_4_1_3_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_1"><i><font color="Green" title="E108">A94</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E5:79_2_1_1_2_1_1_1_4_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E6:79_2_1_1_2_1_1_1_4_1_3_1_1"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1"><i><font color="Green" title="E78">A64</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1"><i><font color="Green" title="E79">A65</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_2"><b>case </b></a><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_2"/><i><font color="Green" title="E107">A95</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E108">A96</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_2"/><i><font color="Green" title="E109">A97</font></i>: 
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E107">A95</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_2_1_1_1_4_1_3_1_2"/><b>then </b><i><font color="Green" title="E110">A98</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E107">A95</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><div><i><font color="Green" title="E111">A99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_2_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A97</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A96</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A97</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_2_2"/>
<font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 ;<br/><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A96</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A97</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E110">A98</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:79_2_1_1_2_1_1_1_4_1_3_1_2"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E108">A96</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E109">A97</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1_3_1_2"><i><font color="Green" title="E111">A99</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:79_2_1_1_2_1_1_1_4_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:79_2_1_1_2_1_1_1_4_1_3_1_2"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1"><i><font color="Green" title="E78">A64</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1"><i><font color="Green" title="E79">A65</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_3"><b>case </b></a><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E107">A100</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_3"/><b>then </b><i><font color="Green" title="E108">A101</font></i>: 
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E109">A102</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E107">A100</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_2_1_1_1_4_1_3_1_3"/><i><font color="Green" title="E110">A103</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E107">A100</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A101</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><div><i><font color="Green" title="E111">A104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_3_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_3_1"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A102</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A101</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E110">A103</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_3_2"/>
<font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 ;<br/><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_3_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A101</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A101</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A102</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/></div><b>end;</b></div><a NAME="E7:79_2_1_1_2_1_1_1_4_1_3_1_3"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E108">A101</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E109">A102</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1_3_1_3"><i><font color="Green" title="E111">A104</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E8:79_2_1_1_2_1_1_1_4_1_3_1_3"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E9:79_2_1_1_2_1_1_1_4_1_3_1_3"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1"><i><font color="Green" title="E78">A64</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1"><i><font color="Green" title="E79">A65</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_2_1_1_2_1_1_1_4_1_3_1_4"><b>case </b></a><a NAME="E1:79_2_1_1_2_1_1_1_4_1_3_1_4"/><i><font color="Green" title="E107">A105</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:79_2_1_1_2_1_1_1_4_1_3_1_4"/><b>then </b><i><font color="Green" title="E108">A106</font></i>: 
( <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:79_2_1_1_2_1_1_1_4_1_3_1_4"/>
( <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c24">gr2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79"><i><font color="Green" title="E68">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1_3_1_4"><i><font color="Green" title="E107">A105</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E4:79_2_1_1_2_1_1_1_4_1_3_1_4"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">gr1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E102">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E103">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2_1_1_1_4_1"><i><font color="Green" title="E104">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1_1_4_1_3_1_4"><i><font color="Green" title="E108">A106</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><a NAME="E5:79_2_1_1_2_1_1_1_4_1_3_1_4"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">r2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E6:79_2_1_1_2_1_1_1_4_1_3_1_4"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:79_2_1_1_2_1_1"><i><font color="Green" title="E78">A64</font></i></a>, <a class="txt" href="jgraph_6.html#E3:79_2_1_1_2_1_1"><i><font color="Green" title="E79">A65</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:79_2_1_1_2_1_1_1_4_1"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_1_1_2_1_1_1_4"/>
<font color="Maroon" title="c15">k</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:79_2_1_1_2_1_1_1"/><i><font color="Green" title="E102">A107</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E25:79_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">k</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1"><i><font color="Green" title="E77">A63</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E26:79_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">rxc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s1</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:79_2_1_1_2_1_1_1"><i><font color="Green" title="E94">A80</font></i></a>, <a class="txt" href="jgraph_6.html#E20:79_2_1_1_2_1_1_1"><i><font color="Green" title="E100">A86</font></i></a>, <a class="txt" href="jgraph_6.html#E23:79_2_1_1_2_1_1_1"><i><font color="Green" title="E101">A87</font></i></a>, <a class="txt" href="jgraph_6.html#E24:79_2_1_1_2_1_1_1"><i><font color="Green" title="E102">A107</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E27:79_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2_1_1_1"><i><font color="Green" title="E87">A73</font></i></a>, <a class="txt" href="jgraph_6.html#E20:79_2_1_1_2_1_1_1"><i><font color="Green" title="E100">A86</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:79_2_1_1_2_1_1"/>
<font color="Maroon" title="c9">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:79_2_1_1_2_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:79_2_1_1_2"><i><font color="Green" title="E75">A61</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2"><i><font color="Green" title="E76">A62</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:79_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:79_2_1_1_2"><i><font color="Green" title="E72">A58</font></i></a>, <a class="txt" href="jgraph_6.html#E4:79_2_1_1_2"><i><font color="Green" title="E75">A61</font></i></a>, <a class="txt" href="jgraph_6.html#E5:79_2_1_1_2"><i><font color="Green" title="E76">A62</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:79_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>
 ;<br/>


</div><b>end;</b></div>


</div>
