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

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p4</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>let </b><font color="Maroon" title="c5">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>let </b><font color="Maroon" title="c6">f</font> be   <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>,<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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1;<br/>
<b>assume </b><a NAME="E1:89"/><i><font color="Green" title="E78">A1</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jgraph_6.html#K5" title="JGRAPH_6:func.5">circle</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="jgraph_3.html#K1" title="JGRAPH_3:func.1">Sq_Circ</a>  )
 ;<br/>

<a NAME="E2:89"/><i><font color="Green" title="E79">A2</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <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="E3:89"/><i><font color="Green" title="E80">A3</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a></i>;<br/>
<a NAME="E4:89"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 </span>}</span> 
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T33" target="_self" title="JGRAPH_6:th.33">Th33</a></i>;<br/>
<b>thus </b><a NAME="E5:89"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 implies <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green" title="E82">A5</font></i>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p4</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 )
 ;<br/>

<a NAME="E2:89_1"/><b>then </b><i><font color="Green" title="E83">A6</font></i>: 
( <font color="Maroon" title="c1">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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp; <font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:89"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/>
<a NAME="E3:89_1"/><i><font color="Green" title="E84">A7</font></i>: 
( <font color="Maroon" title="c3">p3</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 &amp; <font color="Maroon" title="c4">p4</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E2:89"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">q8</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><b> such that </b><br/><a NAME="E5:89_1"/><i><font color="Green" title="E85">A8</font></i>: 
( <font color="Maroon" title="c8">q8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font> &amp; ( ( <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c8">q8</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> 1 ) or ( <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c8">q8</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> 1 ) or ( <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c8">q8</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> 1 ) or ( <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">q8</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c8">q8</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> 1 ) ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89"><i><font color="Green" title="E80">A3</font></i></a></i>;<br/>
<a NAME="E6:89_1"/><i><font color="Green" title="E86">A9</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/>
<a NAME="E7:89_1"/><i><font color="Green" title="E87">A10</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">p4</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/>
<a NAME="E8:89_1"/><i><font color="Green" title="E88">A11</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/>
<a NAME="E9:89_1"/><i><font color="Green" title="E89">A12</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<i><font color="Green" title="E90">A13</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T62" title="EUCLID:th.62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T62" title="EUCLID:th.62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60" title="EUCLID:th.60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>

;<br/>
<a NAME="E11:89_1"/><b>then </b><i><font color="Green" title="E91">A14</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="ref" href="goboard7.html#T7" title="GOBOARD7:th.7">GOBOARD7:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1"/>
( <font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c1">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c1">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1"/><i><font color="Green" title="E92">A15</font></i>: 
<font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1"/><b>then </b><i><font color="Green" title="E93">A16</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p1</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1"/><b>then </b><i><font color="Green" title="E94">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><a NAME="E4:89_1_2_1_1"/><i><font color="Green" title="E95">A18</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E5:89_1_2_1_1"/><i><font color="Green" title="E96">A19</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:89_1_2_1_1"/><b>then </b><i><font color="Green" title="E97">A20</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1"><i><font color="Green" title="E95">A18</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_1"/><i><font color="Green" title="E98">A21</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E99">A22</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="ref" href="jgraph_6.html#T75" target="_self" title="JGRAPH_6:th.75">Th75</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_1"/><i><font color="Green" title="E100">A23</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1"><i><font color="Green" title="E86">A9</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_1"><i><font color="Green" title="E98">A21</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><b>hence </b><a NAME="E5:89_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_1"><i><font color="Green" title="E99">A22</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_1"><i><font color="Green" title="E100">A23</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2"/><i><font color="Green" title="E98">A24</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or  not <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E99">A25</font></i>: 
( <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E100">A26</font></i>: 
( <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">p2</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E101">A27</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1"><i><font color="Green" title="E95">A18</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E102">A28</font></i>: 
<font color="Maroon" title="c1">p1</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="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self" title="JGRAPH_6:th.65">Th65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or  not <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E103">A29</font></i>: 
( <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E104">A30</font></i>: 
( <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c3">p3</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E105">A31</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1"><i><font color="Green" title="E95">A18</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E106">A32</font></i>: 
<font color="Maroon" title="c2">p2</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="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self" title="JGRAPH_6:th.65">Th65</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> ) or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E107">A33</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E108">A34</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E109">A35</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E100">A26</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E110">A36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E104">A30</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E111">A37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E107">A33</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E108">A34</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E102">A28</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E112">A38</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1"><i><font color="Green" title="E94">A17</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1"><i><font color="Green" title="E97">A20</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E109">A35</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E106">A32</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E113">A39</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E105">A31</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E109">A35</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E110">A36</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E114">A40</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1"><i><font color="Green" title="E95">A18</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1"><i><font color="Green" title="E96">A19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E11:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">p3</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E107">A33</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self" title="JGRAPH_6:th.65">Th65</a></i>;<br/><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E107">A33</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E13:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E105">A31</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E110">A36</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E111">A37</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E114">A40</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><b>hence </b><a NAME="E14:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E112">A38</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E113">A39</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E107">A41</font></i>: 
(  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green" title="E108">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E107">A41</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_3"/>
( ( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E109">A43</font></i>: 
<font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E110">A44</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><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="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/><i><font color="Green" title="E111">A45</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E110">A44</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E107">A41</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E110">A44</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4_1"><i><font color="Green" title="E111">A45</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"/>
( ( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E109">A43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E109">A46</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E100">A26</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E110">A47</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E104">A30</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E102">A28</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E111">A48</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1"><i><font color="Green" title="E94">A17</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1"><i><font color="Green" title="E97">A20</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E109">A46</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E103">A29</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E106">A32</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E112">A49</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1"><i><font color="Green" title="E105">A31</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E109">A46</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E110">A47</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><div><i><font color="Green" title="E113">A50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1"/>
( ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E114">A51</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E108">A42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E114">A51</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E114">A51</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"/>
<font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E114">A51</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green" title="E115">A52</font></i>: 
( <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c4">p4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E116">A53</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E114">A51</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E115">A52</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E116">A53</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E114">A54</font></i>: 
(  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E115">A55</font></i>: 
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><i><font color="Green" title="E114">A54</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E116">A56</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E117">A57</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E118">A58</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E119">A59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E120">A60</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E121">A61</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E117">A57</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E118">A58</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E122">A62</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E119">A59</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E121">A61</font></i></a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E123">A63</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E124">A64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E118">A58</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E123">A63</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"><i><font color="Green" title="E115">A55</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E116">A56</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E125">A65</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green" title="E126">A66</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E117">A57</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E118">A58</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E125">A65</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E120">A60</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E126">A66</font></i></a></i>;<br/><b>hence </b><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E122">A62</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E124">A64</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="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E125">A67</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E126">A68</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E127">A69</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E128">A70</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E126">A68</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E129">A71</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E126">A68</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E127">A69</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E130">A72</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E131">A73</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E132">A74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E127">A69</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E131">A73</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E133">A75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E125">A67</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E134">A76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E135">A77</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E136">A78</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E137">A79</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E138">A80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E127">A69</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E137">A79</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E139">A81</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E128">A70</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E134">A76</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E136">A78</font></i></a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E129">A71</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E135">A77</font></i></a></i>;<br/><b>hence </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E138">A80</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E139">A81</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="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E128">A70</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E129">A71</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E130">A72</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E132">A74</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E133">A75</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E125">A82</font></i>: 
<font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E126">A83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E127">A84</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E117">A57</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E118">A58</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E128">A85</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E125">A82</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self" title="JGRAPH_6:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E121">A61</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E124">A64</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E126">A83</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E127">A84</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E128">A85</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E116">A86</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E117">A87</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E118">A88</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E119">A89</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E117">A87</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E120">A90</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E117">A87</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E118">A88</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E121">A91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E116">A86</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E122">A92</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E123">A93</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E124">A94</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E125">A95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E118">A88</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E124">A94</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E126">A96</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E119">A89</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E121">A91</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E123">A93</font></i></a></i>;<br/><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E13:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E120">A90</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E122">A92</font></i></a></i>;<br/><b>hence </b><a NAME="E14:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E125">A95</font></i></a>, <a class="txt" href="jgraph_6.html#E11:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E126">A96</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="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E116">A97</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E117">A98</font></i>: 
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self" title="JGRAPH_6:th.10">Th10</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E118">A99</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E119">A100</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E120">A101</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E118">A99</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E121">A102</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E118">A99</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E119">A100</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E122">A103</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E123">A104</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E116">A97</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E117">A98</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E116">A97</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E120">A101</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E121">A102</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E122">A103</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E123">A104</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E114">A105</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a></i>;<br/><b>thus </b><a NAME="E11:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/><i><font color="Green" title="E115">A106</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E114">A105</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : ( ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_1"><i><font color="Green" title="E115">A106</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/><i><font color="Green" title="E115">A107</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E114">A105</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2_3_2"><i><font color="Green" title="E115">A107</font></i></a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E111">A48</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E112">A49</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E113">A50</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E103">A108</font></i>: 
(  not <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green" title="E104">A109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E103">A108</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_3"/>
( ( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E105">A110</font></i>: 
<font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E106">A111</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><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="89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/><i><font color="Green" title="E107">A112</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E106">A111</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E103">A108</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E106">A111</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4_1"><i><font color="Green" title="E107">A112</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"/>
( ( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E105">A110</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E105">A113</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:89_1"><i><font color="Green" title="E91">A14</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E106">A114</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E100">A26</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E99">A25</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E102">A28</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E107">A115</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1"><i><font color="Green" title="E94">A17</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1"><i><font color="Green" title="E97">A20</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1"><i><font color="Green" title="E101">A27</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E106">A114</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E108">A116</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1"><i><font color="Green" title="E89">A12</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E105">A113</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a>, <a class="ref" href="goboard7.html#T7" title="GOBOARD7:th.7">GOBOARD7:7</a></i>;<br/><div><i><font color="Green" title="E109">A117</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1"/>
( ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E110">A118</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E111">A119</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E112">A120</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E113">A121</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E114">A122</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T28" target="_self" title="JGRAPH_6:th.28">Th28</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E115">A123</font></i>: 
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E111">A119</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E112">A120</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E113">A121</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E114">A122</font></i></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E104">A109</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E116">A124</font></i>: 
( <font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
 <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="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E115">A123</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_1"><i><font color="Green" title="E116">A124</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self" title="JGRAPH_6:th.65">Th65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><i><font color="Green" title="E116">A125</font></i>: 
<font color="Maroon" title="c3">p3</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E115">A123</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_2"><i><font color="Green" title="E116">A125</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><i><font color="Green" title="E116">A126</font></i>: 
<font color="Maroon" title="c3">p3</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E115">A123</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_3"><i><font color="Green" title="E116">A126</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green" title="E116">A127</font></i>: 
( <font color="Maroon" title="c3">p3</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c3">p3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E115">A123</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E117">A128</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><i><font color="Green" title="E118">A129</font></i>: 
<font color="Maroon" title="c3">p3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E116">A127</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"><i><font color="Green" title="E110">A118</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E116">A127</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E117">A128</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E118">A129</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E110">A130</font></i>: 
(  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/><i><font color="Green" title="E111">A131</font></i>: 
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><i><font color="Green" title="E110">A130</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E112">A132</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E113">A133</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E114">A134</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E115">A135</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E116">A136</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E117">A137</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E113">A133</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E114">A134</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E118">A138</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E115">A135</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E117">A137</font></i></a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E119">A139</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E120">A140</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E114">A134</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E119">A139</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"><i><font color="Green" title="E111">A131</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E112">A132</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E121">A141</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green" title="E122">A142</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E113">A133</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E114">A134</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E121">A141</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E116">A136</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E122">A142</font></i></a></i>;<br/><b>hence </b><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E118">A138</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E120">A140</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="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E121">A143</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E122">A144</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E123">A145</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E124">A146</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E122">A144</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E125">A147</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E122">A144</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E123">A145</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E126">A148</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E127">A149</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E128">A150</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E123">A145</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E127">A149</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E129">A151</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E121">A143</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E130">A152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E131">A153</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E132">A154</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E133">A155</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E134">A156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E123">A145</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E133">A155</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E135">A157</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E124">A146</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E130">A152</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E132">A154</font></i></a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E125">A147</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E131">A153</font></i></a></i>;<br/><b>hence </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E134">A156</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E135">A157</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="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E124">A146</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E125">A147</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E126">A148</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E128">A150</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E129">A151</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E121">A158</font></i>: 
<font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E122">A159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E123">A160</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E113">A133</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E114">A134</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E124">A161</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E121">A158</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self" title="JGRAPH_6:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E117">A137</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E120">A140</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E122">A159</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E123">A160</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E124">A161</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E112">A162</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E113">A163</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E114">A164</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E115">A165</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E113">A163</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E116">A166</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E113">A163</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E114">A164</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E117">A167</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E112">A162</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E118">A168</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E119">A169</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E120">A170</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E121">A171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E114">A164</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E120">A170</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="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E122">A172</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E115">A165</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E117">A167</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E119">A169</font></i></a></i>;<br/><a NAME="E12:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E13:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E116">A166</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E118">A168</font></i></a></i>;<br/><b>hence </b><a NAME="E14:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E121">A171</font></i></a>, <a class="txt" href="jgraph_6.html#E11:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E122">A172</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="89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E112">A173</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E113">A174</font></i>: 
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self" title="JGRAPH_6:th.10">Th10</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E114">A175</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E115">A176</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E116">A177</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E114">A175</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E117">A178</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E114">A175</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E115">A176</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E118">A179</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E119">A180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E112">A173</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E113">A174</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E112">A173</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E116">A177</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E117">A178</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E118">A179</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E119">A180</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E110">A181</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a></i>;<br/><b>thus </b><a NAME="E10:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/><i><font color="Green" title="E111">A182</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E110">A181</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : ( ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_1"><i><font color="Green" title="E111">A182</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/><i><font color="Green" title="E111">A183</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E110">A181</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_1_1_1_2_3_2"><i><font color="Green" title="E111">A183</font></i></a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E11:89_1_2_1_1_1_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E107">A115</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E108">A116</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_1_1_1_2"><i><font color="Green" title="E109">A117</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:89_1_2_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E99">A184</font></i>: 
(  not <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 ;<br/><div class="add"><div><i><font color="Green" title="E100">A185</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_1_1_1"/>
( ( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E99">A184</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_1_1_2"/>
( ( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_1_1_3"/>
( ( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green" title="E101">A186</font></i>: 
<font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_1_1_4"/><i><font color="Green" title="E102">A187</font></i>: 
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><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="89_1_2_1_1_1_1_2_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/><i><font color="Green" title="E103">A188</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green" title="E102">A187</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E99">A184</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green" title="E102">A187</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_1_1_4_1"><i><font color="Green" title="E103">A188</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_1_1_4"/>
( ( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_1_1_4"><i><font color="Green" title="E101">A186</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E101">A189</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11:89_1"><i><font color="Green" title="E91">A14</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E102">A190</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1"><i><font color="Green" title="E89">A12</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a>, <a class="ref" href="goboard7.html#T7" title="GOBOARD7:th.7">GOBOARD7:7</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E101">A189</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E103">A191</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1"><i><font color="Green" title="E89">A12</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1"><i><font color="Green" title="E90">A13</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a>, <a class="ref" href="goboard7.html#T7" title="GOBOARD7:th.7">GOBOARD7:7</a></i>;<br/><div><i><font color="Green" title="E104">A192</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1"/>
( ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E105">A193</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E106">A194</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E107">A195</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E108">A196</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E109">A197</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T28" target="_self" title="JGRAPH_6:th.28">Th28</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E110">A198</font></i>: 
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E105">A193</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E106">A194</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E107">A195</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E108">A196</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E109">A197</font></i></a>, <a class="ref" href="goboard7.html#T8" title="GOBOARD7:th.8">GOBOARD7:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1"/>
( ( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or ( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E100">A185</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E111">A199</font></i>: 
( <font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/><b>then </b>
 <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="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E105">A193</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E110">A198</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_1"><i><font color="Green" title="E111">A199</font></i></a>, <a class="ref" href="jgraph_6.html#T65" target="_self" title="JGRAPH_6:th.65">Th65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><i><font color="Green" title="E111">A200</font></i>: 
<font color="Maroon" title="c2">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E110">A198</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E105">A193</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_2"><i><font color="Green" title="E111">A200</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><i><font color="Green" title="E111">A201</font></i>: 
<font color="Maroon" title="c2">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E110">A198</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E105">A193</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_3"><i><font color="Green" title="E111">A201</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><i><font color="Green" title="E111">A202</font></i>: 
( <font color="Maroon" title="c2">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E110">A198</font></i></a>, <a class="ref" href="jgraph_6.html#T69" target="_self" title="JGRAPH_6:th.69">Th69</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E112">A203</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p4</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><i><font color="Green" title="E113">A204</font></i>: 
<font color="Maroon" title="c2">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E111">A202</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1"><i><font color="Green" title="E105">A193</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E111">A202</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E112">A203</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_1_1_1_4"><i><font color="Green" title="E113">A204</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E105">A205</font></i>: 
(  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or  not <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c1">p1</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">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2"/><i><font color="Green" title="E106">A206</font></i>: 
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="jgraph_6.html#T73" target="_self" title="JGRAPH_6:th.73">Th73</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E105">A205</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E107">A207</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E108">A208</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E109">A209</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E110">A210</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E111">A211</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E112">A212</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E108">A208</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E109">A209</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E113">A213</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E110">A210</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E112">A212</font></i></a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/><i><font color="Green" title="E114">A214</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E115">A215</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E109">A209</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E114">A214</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="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2"><i><font color="Green" title="E106">A206</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E107">A207</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E116">A216</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><i><font color="Green" title="E117">A217</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E108">A208</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E109">A209</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E116">A216</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E111">A211</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"><i><font color="Green" title="E117">A217</font></i></a></i>;<br/><b>hence </b><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E113">A213</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E115">A215</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="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">p4</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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E116">A218</font></i>: 
<font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E117">A219</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E118">A220</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E119">A221</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E117">A219</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E120">A222</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E117">A219</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E118">A220</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E121">A223</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E122">A224</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E123">A225</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E118">A220</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E122">A224</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="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E124">A226</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E116">A218</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E125">A227</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E126">A228</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E127">A229</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E128">A230</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E129">A231</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E118">A220</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E128">A230</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="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><i><font color="Green" title="E130">A232</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E119">A221</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E125">A227</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E127">A229</font></i></a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E120">A222</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E126">A228</font></i></a></i>;<br/><b>hence </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E129">A231</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_1"><i><font color="Green" title="E130">A232</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="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E119">A221</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E120">A222</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E121">A223</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E123">A225</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"><i><font color="Green" title="E124">A226</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E116">A233</font></i>: 
<font color="Maroon" title="c4">p4</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p4</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> 1 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E117">A234</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E118">A235</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E108">A208</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E109">A209</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/><i><font color="Green" title="E119">A236</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E116">A233</font></i></a>, <a class="ref" href="jgraph_6.html#T80" target="_self" title="JGRAPH_6:th.80">Th80</a></i>;<br/><b>hence </b><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E112">A212</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"><i><font color="Green" title="E115">A215</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E117">A234</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E118">A235</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1_2_1_3"><i><font color="Green" title="E119">A236</font></i></a>, <a class="ref" href="jgraph_5.html#T59" title="JGRAPH_5:th.59">JGRAPH_5:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E107">A237</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E108">A238</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E109">A239</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E110">A240</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E108">A238</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E111">A241</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E108">A238</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E109">A239</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E112">A242</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E107">A237</font></i></a>, <a class="ref" href="jgraph_6.html#T79" target="_self" title="JGRAPH_6:th.79">Th79</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E113">A243</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E114">A244</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T37" title="JGRAPH_5:th.37">JGRAPH_5:37</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E115">A245</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_6.html#T18" target="_self" title="JGRAPH_6:th.18">Th18</a>, <a class="ref" href="jgraph_5.html#T32" title="JGRAPH_5:th.32">JGRAPH_5:32</a></i>;<br/><div><i><font color="Green" title="E116">A246</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p1</font> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E109">A239</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E115">A245</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="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><a NAME="E11:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><i><font color="Green" title="E117">A247</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E110">A240</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E112">A242</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E114">A244</font></i></a></i>;<br/><a NAME="E12:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Olive" title="b1">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="jgraph_5.html#T38" title="JGRAPH_5:th.38">JGRAPH_5:38</a></i>;<br/><a NAME="E13:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E111">A241</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E113">A243</font></i></a></i>;<br/><b>hence </b><a NAME="E14:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E116">A246</font></i></a>, <a class="txt" href="jgraph_6.html#E11:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_2"><i><font color="Green" title="E117">A247</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="89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E107">A248</font></i>: 
( <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">p4</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E108">A249</font></i>: 
<font color="Maroon" title="c4">p4</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1"><i><font color="Green" title="E85">A8</font></i></a>, <a class="ref" href="jgraph_6.html#T10" target="_self" title="JGRAPH_6:th.10">Th10</a></i>;<br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E109">A250</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a>, <a class="ref" href="jgraph_6.html#T45" target="_self" title="JGRAPH_6:th.45">Th45</a>, <a class="ref" href="jgraph_3.html#T33" title="JGRAPH_3:th.33">JGRAPH_3:33</a></i>;<br/><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E110">A251</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</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 <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#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E111">A252</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E109">A250</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E112">A253</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E109">A250</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E110">A251</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E113">A254</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1"><i><font color="Green" title="E93">A16</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="ref" href="jgraph_6.html#T77" target="_self" title="JGRAPH_6:th.77">Th77</a></i>;<br/><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/><i><font color="Green" title="E114">A255</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1"><i><font color="Green" title="E92">A15</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2"><i><font color="Green" title="E98">A24</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E107">A248</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E108">A249</font></i></a>, <a class="ref" href="jgraph_6.html#T81" target="_self" title="JGRAPH_6:th.81">Th81</a></i>;<br/><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E107">A248</font></i></a>, <a class="ref" href="jgraph_6.html#T78" target="_self" title="JGRAPH_6:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E10:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E111">A252</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E112">A253</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E113">A254</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2_2_1_2_1_1_3"><i><font color="Green" title="E114">A255</font></i></a>, <a class="ref" href="jgraph_5.html#T54" title="JGRAPH_5:th.54">JGRAPH_5:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:89_1_2_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E105">A256</font></i>: 
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E35"><i><font color="Green" title="E26">Lm8</font></i></a></i>;<br/><b>thus </b><a NAME="E9:89_1_2_1_1_1_1_2_1_1_2"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_3"/>
 <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_3_1"/><i><font color="Green" title="E106">A257</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E105">A256</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : ( ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">q</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> 1 ) ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_3_1"><i><font color="Green" title="E106">A257</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:89_1_2_1_1_1_1_2_1_1_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_2_1_1_1_1_2_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_1_2_1_1_1_1_2_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : ( ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Olive" title="b1">p</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> 1 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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><b> such that </b><br/><a NAME="E3:89_1_2_1_1_1_1_2_1_1_2_3_2"/><i><font color="Green" title="E106">A258</font></i>: 
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; ( ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or (  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) or ( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c10">p</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> 1 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:89_1_2_1_1_1_1_2_1_1_2_3_2"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E105">A256</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_1_1_1_2_1_1_2_3_2"><i><font color="Green" title="E106">A258</font></i></a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E10:89_1_2_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E4:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E102">A190</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E103">A191</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_1_1_1_2_1_1_2"><i><font color="Green" title="E104">A192</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:89_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:89_1_2_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_2"><b>case </b></a><a NAME="E1:89_1_2_1_2"/><i><font color="Green" title="E92">A259</font></i>: 
<font color="Maroon" title="c1">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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_2"/><i><font color="Green" title="E93">A260</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E3:89_1_2_1_2"/><i><font color="Green" title="E94">A261</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E4:89_1_2_1_2"/><i><font color="Green" title="E95">A262</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:89_1_2_1_2"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">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> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_2"><i><font color="Green" title="E92">A259</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E6:89_1_2_1_2"/><b>then </b><i><font color="Green" title="E96">A263</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c1">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_2"><i><font color="Green" title="E92">A259</font></i></a>, <a class="txt" href="jgraph_6.html#E2:89_1_2_1_2"><i><font color="Green" title="E93">A260</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_2"><i><font color="Green" title="E95">A262</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self" title="JGRAPH_6:th.70">Th70</a></i>;<br/><a NAME="E7:89_1_2_1_2"/><b>then </b><i><font color="Green" title="E97">A264</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_2"><i><font color="Green" title="E94">A261</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_2"><i><font color="Green" title="E95">A262</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E8:89_1_2_1_2"/><i><font color="Green" title="E98">A265</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_2"><i><font color="Green" title="E96">A263</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E9:89_1_2_1_2"/><b>then </b><i><font color="Green" title="E99">A266</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_2"><i><font color="Green" title="E94">A261</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_2"><i><font color="Green" title="E95">A262</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E10:89_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_2"><i><font color="Green" title="E98">A265</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E11:89_1_2_1_2"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_2"><i><font color="Green" title="E94">A261</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_2"><i><font color="Green" title="E95">A262</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><b>hence </b><a NAME="E12:89_1_2_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_2"><i><font color="Green" title="E97">A264</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_2"><i><font color="Green" title="E99">A266</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_3"><b>case </b></a><a NAME="E1:89_1_2_1_3"/><i><font color="Green" title="E92">A267</font></i>: 
<font color="Maroon" title="c1">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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_3"/><i><font color="Green" title="E93">A268</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E3:89_1_2_1_3"/><i><font color="Green" title="E94">A269</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E4:89_1_2_1_3"/><i><font color="Green" title="E95">A270</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:89_1_2_1_3"/><i><font color="Green" title="E96">A271</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E6:89_1_2_1_3"/><i><font color="Green" title="E97">A272</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E7:89_1_2_1_3"/><i><font color="Green" title="E98">A273</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E8:89_1_2_1_3"/><b>then </b><i><font color="Green" title="E99">A274</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_3"><i><font color="Green" title="E93">A268</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_3"><i><font color="Green" title="E95">A270</font></i></a>, <a class="txt" href="jgraph_6.html#E6:89_1_2_1_3"><i><font color="Green" title="E97">A272</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self" title="JGRAPH_6:th.70">Th70</a></i>;<br/><a NAME="E9:89_1_2_1_3"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c1">p1</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> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_3"><i><font color="Green" title="E92">A267</font></i></a>, <a class="ref" href="jgraph_6.html#T9" target="_self" title="JGRAPH_6:th.9">Th9</a></i>;<br/><a NAME="E10:89_1_2_1_3"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c1">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_3"><i><font color="Green" title="E92">A267</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_3"><i><font color="Green" title="E96">A271</font></i></a>, <a class="txt" href="jgraph_6.html#E7:89_1_2_1_3"><i><font color="Green" title="E98">A273</font></i></a>, <a class="ref" href="jgraph_6.html#T71" target="_self" title="JGRAPH_6:th.71">Th71</a></i>;<br/><a NAME="E11:89_1_2_1_3"/><b>then </b><i><font color="Green" title="E100">A275</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c1">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E8:89_1_2_1_3"><i><font color="Green" title="E99">A274</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E12:89_1_2_1_3"/><b>then </b><i><font color="Green" title="E101">A276</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_3"><i><font color="Green" title="E94">A269</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_3"><i><font color="Green" title="E95">A270</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E13:89_1_2_1_3"/><i><font color="Green" title="E102">A277</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E11:89_1_2_1_3"><i><font color="Green" title="E100">A275</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E14:89_1_2_1_3"/><b>then </b><i><font color="Green" title="E103">A278</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_3"><i><font color="Green" title="E94">A269</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_3"><i><font color="Green" title="E95">A270</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E15:89_1_2_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E13:89_1_2_1_3"><i><font color="Green" title="E102">A277</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E16:89_1_2_1_3"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_3"><i><font color="Green" title="E94">A269</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_3"><i><font color="Green" title="E95">A270</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><b>hence </b><a NAME="E17:89_1_2_1_3"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E12:89_1_2_1_3"><i><font color="Green" title="E101">A276</font></i></a>, <a class="txt" href="jgraph_6.html#E14:89_1_2_1_3"><i><font color="Green" title="E103">A278</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2_1_4"><b>case </b></a><a NAME="E1:89_1_2_1_4"/><i><font color="Green" title="E92">A279</font></i>: 
( <font color="Maroon" title="c1">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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:89_1_2_1_4"/><i><font color="Green" title="E93">A280</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E3:89_1_2_1_4"/><i><font color="Green" title="E94">A281</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E4:89_1_2_1_4"/><i><font color="Green" title="E95">A282</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E5:89_1_2_1_4"/><i><font color="Green" title="E96">A283</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E6:89_1_2_1_4"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E7:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E97">A284</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E2:89_1_2_1_4"><i><font color="Green" title="E93">A280</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_4"><i><font color="Green" title="E95">A282</font></i></a>, <a class="txt" href="jgraph_6.html#E5:89_1_2_1_4"><i><font color="Green" title="E96">A283</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self" title="JGRAPH_6:th.70">Th70</a></i>;<br/><a NAME="E8:89_1_2_1_4"/><i><font color="Green" title="E98">A285</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E9:89_1_2_1_4"/><i><font color="Green" title="E99">A286</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E10:89_1_2_1_4"/><i><font color="Green" title="E100">A287</font></i>: 
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E11:89_1_2_1_4"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E5:89_1_2_1_4"><i><font color="Green" title="E96">A283</font></i></a>, <a class="txt" href="jgraph_6.html#E8:89_1_2_1_4"><i><font color="Green" title="E98">A285</font></i></a>, <a class="ref" href="jgraph_6.html#T70" target="_self" title="JGRAPH_6:th.70">Th70</a></i>;<br/><a NAME="E12:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E101">A288</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E7:89_1_2_1_4"><i><font color="Green" title="E97">A284</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E13:89_1_2_1_4"/>
 <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</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"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T56" target="_self" title="JGRAPH_6:th.56">Th56</a></i>;<br/><a NAME="E14:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E102">A289</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E15:89_1_2_1_4"/>
( <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">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> 1 )
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_4"><i><font color="Green" title="E92">A279</font></i></a>, <a class="ref" href="jgraph_6.html#T11" target="_self" title="JGRAPH_6:th.11">Th11</a></i>;<br/><a NAME="E16:89_1_2_1_4"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c1">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1_2_1_4"><i><font color="Green" title="E92">A279</font></i></a>, <a class="txt" href="jgraph_6.html#E9:89_1_2_1_4"><i><font color="Green" title="E99">A286</font></i></a>, <a class="txt" href="jgraph_6.html#E10:89_1_2_1_4"><i><font color="Green" title="E100">A287</font></i></a>, <a class="txt" href="jgraph_6.html#E14:89_1_2_1_4"><i><font color="Green" title="E102">A289</font></i></a>, <a class="ref" href="jgraph_6.html#T72" target="_self" title="JGRAPH_6:th.72">Th72</a></i>;<br/><a NAME="E17:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E103">A290</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c1">p1</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E12:89_1_2_1_4"><i><font color="Green" title="E101">A288</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E18:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E104">A291</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_4"><i><font color="Green" title="E94">A281</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_4"><i><font color="Green" title="E95">A282</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E19:89_1_2_1_4"/><i><font color="Green" title="E105">A292</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c2">p2</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E17:89_1_2_1_4"><i><font color="Green" title="E103">A290</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E20:89_1_2_1_4"/><b>then </b><i><font color="Green" title="E106">A293</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_4"><i><font color="Green" title="E94">A281</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_4"><i><font color="Green" title="E95">A282</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><a NAME="E21:89_1_2_1_4"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<font color="Maroon" title="c3">p3</font>, <a href="jgraph_6.html#NK1" title="JGRAPH_6:NK.1">rectangle</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1,<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E19:89_1_2_1_4"><i><font color="Green" title="E105">A292</font></i></a>, <a class="ref" href="jgraph_6.html#T60" target="_self" title="JGRAPH_6:th.60">Th60</a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a></i>;<br/><a NAME="E22:89_1_2_1_4"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font>,<font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:89"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:89_1"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E3:89_1_2_1_4"><i><font color="Green" title="E94">A281</font></i></a>, <a class="txt" href="jgraph_6.html#E4:89_1_2_1_4"><i><font color="Green" title="E95">A282</font></i></a>, <a class="ref" href="jgraph_6.html#T76" target="_self" title="JGRAPH_6:th.76">Th76</a></i>;<br/><b>hence </b><a NAME="E23:89_1_2_1_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E18:89_1_2_1_4"><i><font color="Green" title="E104">A291</font></i></a>, <a class="txt" href="jgraph_6.html#E20:89_1_2_1_4"><i><font color="Green" title="E106">A293</font></i></a>, <a class="ref" href="jordan17.html#D1" title="JORDAN17:def.1">JORDAN17:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:89_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p4</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c5">P</font>
 ;<br/>


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


</div>
