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

<b>let </b><font color="Maroon" title="c1">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>assume </b><a NAME="E1:35"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</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> 
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36" title="JGRAPH_3:th.36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:35"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green" title="E34">A4</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">P2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><b> such that </b><br/><a NAME="E6:35"/><i><font color="Green" title="E35">A5</font></i>: 
( <font color="Maroon" title="c2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <font color="Maroon" title="c2">P2</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">P4</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>;<br/>
<a NAME="E7:35"/><i><font color="Green" title="E36">A6</font></i>: 
(  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E8:35"/><i><font color="Green" title="E37">A7</font></i>: 
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font> <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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<a NAME="E9:35"/><i><font color="Green" title="E38">A8</font></i>: 
 <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">P1</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>;<br/>
<b>set </b><font color="Maroon" title="c5">P2</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>;<br/>
<b>set </b><font color="Maroon" title="c6">Q</font> =  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">p11</font> =  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>;<br/>
<b>set </b><font color="Maroon" title="c8">p22</font> =  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>;<br/>
<b>set </b><font color="Maroon" title="c9">p8</font> =  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">pj</font> =  <a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<a NAME="E10:35"/><i><font color="Green" title="E39">A9</font></i>: 
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">W-bound</a> <font color="Maroon" title="c1">P</font> <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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<a NAME="E11:35"/><i><font color="Green" title="E40">A10</font></i>: 
 <a href="pscomp_1.html#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green" title="E41">A11</font></i>: 
 <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">S-bound</a> <font color="Maroon" title="c1">P</font> <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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<a NAME="E13:35"/><i><font color="Green" title="E42">A12</font></i>: 
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">N-bound</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T31" target="_self" title="JGRAPH_5:th.31">Th31</a></i>;<br/>
<a NAME="E14:35"/><b>then </b><i><font color="Green" title="E43">A13</font></i>: 
 <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E12:35"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="jordan6.html#T84" title="JORDAN6:th.84">JORDAN6:84</a></i>;<br/>
<a NAME="E15:35"/><i><font color="Green" title="E44">A14</font></i>: 
 <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:35_1"/>
<font color="Maroon" title="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<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#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p5">(<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="p3">(<span class="default"><font color="Olive" title="b1">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p4"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">l</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font> &amp; <font color="Olive" title="b1">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">l</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:35_1"/><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><font color="Maroon" title="c12">l</font> <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> &amp; <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<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="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<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#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7" title="TOPREAL3:th.7">TOPREAL3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<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> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<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> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<b>hence </b><a NAME="E6:35_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:35_1"><i><font color="Green" title="E45">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:35"/><b>then </b><i><font color="Green" title="E45">A16</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:35"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E17:35"/><i><font color="Green" title="E46">A17</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<a NAME="E18:35"/>
 <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T33" title="JORDAN6:th.33">JORDAN6:33</a></i>;<br/>
<a NAME="E19:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E17:35"><i><font color="Green" title="E46">A17</font></i></a>, <a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>
<a NAME="E20:35"/><b>then </b><i><font color="Green" title="E47">A18</font></i>: 
(  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b3">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b3">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s2</font> holds <br/> not <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="jordan5c.html#D1" title="JORDAN5C:def.1">JORDAN5C:def 1</a></i>;<br/>
<a NAME="E21:35"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:35_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ;<br/>

<a NAME="E2:35_2"/><b>then </b><i><font color="Green" title="E48">A19</font></i>: 
( <font color="Maroon" title="c11">x</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="c1">P</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">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="E4:35_2"/><i><font color="Green" title="E49">A20</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c12">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<a NAME="E5:35_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_2"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">q</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="E7:35_2"/><i><font color="Green" title="E50">A21</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<a NAME="E8:35_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60" title="SQUARE_1:th.60">SQUARE_1:60</a></i>;<br/>
<a NAME="E9:35_2"/><b>then </b>
( <font color="Maroon" title="c13">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">q</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="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/>
<a NAME="E10:35_2"/><b>then </b>
( <font color="Maroon" title="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35_2"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_2"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E11:35_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:35"/><b>then </b>
(  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E20:35"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E23:35"/><b>then </b><i><font color="Green" title="E48">A22</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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 or <span class="p1">(<span class="default"><a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> 1 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E24:35"/>
 <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="jgraph_5.html#E12:35"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="jgraph_5.html#E13:35"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="jordan6.html#T85" title="JORDAN6:th.85">JORDAN6:85</a></i>;<br/>
<a NAME="E25:35"/><b>then </b><i><font color="Green" title="E49">A23</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E15:35"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="xboole_1.html#T64" title="XBOOLE_1:th.64">XBOOLE_1:64</a></i>;<br/>
<a NAME="E26:35"/><i><font color="Green" title="E50">A24</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<a NAME="E27:35"/>
 <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T33" title="JORDAN6:th.33">JORDAN6:33</a></i>;<br/>
<a NAME="E28:35"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E26:35"><i><font color="Green" title="E50">A24</font></i></a>, <a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>
<a NAME="E29:35"/><b>then </b><i><font color="Green" title="E51">A25</font></i>: 
(  <a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">s2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b2">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/>for <font color="Olive" title="b3">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b3">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">s2</font> holds <br/> not <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="jgraph_5.html#E25:35"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="jordan5c.html#D2" title="JORDAN5C:def.2">JORDAN5C:def 2</a></i>;<br/>
<a NAME="E30:35"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">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="E1:35_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 ;<br/>

<a NAME="E2:35_3"/><b>then </b><i><font color="Green" title="E52">A26</font></i>: 
( <font color="Maroon" title="c11">x</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="c1">P</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">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="E4:35_3"/><i><font color="Green" title="E53">A27</font></i>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c12">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<a NAME="E5:35_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_3"><i><font color="Green" title="E52">A26</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">q</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="E7:35_3"/><i><font color="Green" title="E54">A28</font></i>: 
( <font color="Maroon" title="c13">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/>
<a NAME="E8:35_3"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35_3"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_3"><i><font color="Green" title="E54">A28</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60" title="SQUARE_1:th.60">SQUARE_1:60</a></i>;<br/>
<a NAME="E9:35_3"/><b>then </b>
( <font color="Maroon" title="c13">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">q</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="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/>
<a NAME="E10:35_3"/><b>then </b>
( <font color="Maroon" title="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E4:35_3"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_3"><i><font color="Green" title="E54">A28</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>
<b>hence </b><a NAME="E11:35_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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 class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:35"/><b>then </b><i><font color="Green" title="E52">A29</font></i>: 
(  <a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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  <a href="jordan5c.html#K2" title="JORDAN5C:func.2">Last_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E29:35"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E32:35"/><i><font color="Green" title="E53">A30</font></i>: 
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <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="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E20:35"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E33:35"/><i><font color="Green" title="E54">A31</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E34:35"/><i><font color="Green" title="E55">A32</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E35:35"/>
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E36:35"/><b>then </b><i><font color="Green" title="E56">A33</font></i>: 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</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="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E37:35"/>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E38:35"/><b>then </b><i><font color="Green" title="E57">A34</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</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="c1">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">R</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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>consider </b><font color="Maroon" title="c12">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">R</font></span>)</span><b> such that </b><br/><a NAME="E41:35"/><i><font color="Green" title="E58">A35</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">R</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c11">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c13">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E44:35"/><i><font color="Green" title="E59">A36</font></i>: 
( <font color="Maroon" title="c13">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp;  <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x8</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E32:35"><i><font color="Green" title="E53">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E45:35"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E46:35"/><b>then </b>
<font color="Maroon" title="c13">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <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_5.html#E44:35"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">r8</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E48:35"/><i><font color="Green" title="E60">A37</font></i>: 
( <font color="Maroon" title="c13">x8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r8</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r8</font> &amp; <font color="Maroon" title="c14">r8</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<div><i><font color="Green" title="E61">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_5"/>
<font color="Maroon" title="c14">r8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:35_5"/><b>then </b>
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>,<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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E44:35"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="jgraph_5.html#E48:35"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E3:35_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E23:35"><i><font color="Green" title="E48">A22</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="35_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_6"/>
<font color="Maroon" title="c14">r8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:35_6"/><b>then </b>
 <a href="jordan5c.html#K1" title="JORDAN5C:func.1">First_Point</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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">1,<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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E44:35"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="jgraph_5.html#E48:35"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:35_6"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E23:35"><i><font color="Green" title="E48">A22</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="E51:35"/><b>then </b><i><font color="Green" title="E62">A39</font></i>: 
1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">r8</font>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E48:35"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">h2</font> =  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E53:35"/><i><font color="Green" title="E63">A40</font></i>: 
<font color="Maroon" title="c12">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E54:35"/><i><font color="Green" title="E64">A41</font></i>: 
<font color="Maroon" title="c12">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E55:35"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds  <font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 
;<br/>
<a NAME="E56:35"/><b>then </b><i><font color="Green" title="E65">A42</font></i>: 
<font color="Maroon" title="c15">h2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T35" target="_self" title="JGRAPH_5:th.35">Th35</a></i>;<br/>
<a NAME="E57:35"/><i><font color="Green" title="E66">A43</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E58:35"/><i><font color="Green" title="E67">A44</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E59:35"/><b>then </b><i><font color="Green" title="E68">A45</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E60:35"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h2</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="E61:35"/><b>then </b><i><font color="Green" title="E69">A46</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E57:35"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jgraph_5.html#E59:35"><i><font color="Green" title="E68">A45</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E62:35"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">g0</font> = <font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E61:35"><i><font color="Green" title="E69">A46</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E64:35"/><i><font color="Green" title="E70">A47</font></i>: 
(  ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies for <font color="Olive" title="b1">q</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>  st <font color="Olive" title="b1">q</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="c1">P</font> holds <br/><font color="Olive" title="b1">q</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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_7"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">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>, <font color="Maroon" title="c18">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:35_7"/><i><font color="Green" title="E71">A48</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font> &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p</font> &amp; <font color="Maroon" title="c17">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_7_1"/>
 ex <font color="Olive" title="b1">q</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> st <br/>( <font color="Olive" title="b1">q</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="c1">P</font> &amp; <font color="Olive" title="b1">q</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/><b>then consider </b><font color="Maroon" title="c19">q</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="E3:35_7_1"/><i><font color="Green" title="E72">A49</font></i>: 
( <font color="Maroon" title="c19">q</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="c1">P</font> &amp; <font color="Maroon" title="c19">q</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/> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">R</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c11">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_7_1"/><i><font color="Green" title="E73">A50</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c19">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7_1"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:35_7_1"/><i><font color="Green" title="E74">A51</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:35_7_1"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <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_5.html#E6:35_7_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:35_7_1"/><i><font color="Green" title="E75">A52</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><i><font color="Green" title="E76">A53</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font> = 
<font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">q</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_7_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c19">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E12:35_7_1"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> <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_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a></i>;<br/><a NAME="E13:35_7_1"/><b>then </b><i><font color="Green" title="E77">A54</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E78">A55</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> = 
<font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">p</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c17">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_7_1_4_1"/>
( <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font> or <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r</font> or <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_1"><b>case </b></a><a NAME="E1:35_7_1_4_1_1"/><i><font color="Green" title="E79">A56</font></i>: 
<font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c22">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c18">t</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_7_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E13:35_7_1"><i><font color="Green" title="E77">A54</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">B0</font> = <font color="Maroon" title="c22">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c24">g</font> = <font color="Maroon" title="c16">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c23">B0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">B0</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:35_7_1_4_1_1"/>
<font color="Maroon" title="c16">g0</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E53:35"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jgraph_5.html#E56:35"><i><font color="Green" title="E65">A42</font></i></a>, <a class="ref" href="jgraph_5.html#T10" target="_self" title="JGRAPH_5:th.10">Th10</a></i>;<br/><a NAME="E6:35_7_1_4_1_1"/><b>then </b><i><font color="Green" title="E80">A57</font></i>: 
<font color="Maroon" title="c24">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E7:35_7_1_4_1_1"/><i><font color="Green" title="E81">A58</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_1"><i><font color="Green" title="E79">A56</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/><a NAME="E8:35_7_1_4_1_1"/>
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_1"><i><font color="Green" title="E79">A56</font></i></a></i>;<br/><a NAME="E9:35_7_1_4_1_1"/><b>then </b><i><font color="Green" title="E82">A59</font></i>: 
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:35_7_1_4_1_1"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_1"><i><font color="Green" title="E79">A56</font></i></a></i>;<br/><a NAME="E11:35_7_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:35_7_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <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="c19">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c17">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_7_1"><i><font color="Green" title="E72">A49</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_7_1"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_7_1_4_1_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:35_7_1_4_1_1"/><i><font color="Green" title="E83">A60</font></i>: 
( <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">r1</font> &amp; <font color="Maroon" title="c25">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_1"><i><font color="Green" title="E79">A56</font></i></a>, <a class="txt" href="jgraph_5.html#E6:35_7_1_4_1_1"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_7_1_4_1_1"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="topreal5.html#T14" title="TOPREAL5:th.14">TOPREAL5:14</a></i>;<br/><a NAME="E15:35_7_1_4_1_1"/>
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a></i>;<br/><a NAME="E16:35_7_1_4_1_1"/><b>then </b><i><font color="Green" title="E84">A61</font></i>: 
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:35_7_1_4_1_1"/>
<font color="Maroon" title="c25">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:35_7_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r2</font> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <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_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a></i>;<br/><a NAME="E19:35_7_1_4_1_1"/><b>then </b><i><font color="Green" title="E85">A62</font></i>: 
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:35_7_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:35_7_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E58:35"><i><font color="Green" title="E67">A44</font></i></a></i>;<br/><a NAME="E22:35_7_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E33:35"><i><font color="Green" title="E54">A31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">q3</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="E24:35_7_1_4_1_1"/><i><font color="Green" title="E86">A63</font></i>: 
( <font color="Maroon" title="c26">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c26">q3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/><i><font color="Green" title="E87">A64</font></i>: <font color="Maroon" title="c26">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<font color="Maroon" title="c15">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<font color="Maroon" title="c16">g0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35_7_1_4_1_1"><i><font color="Green" title="E84">A61</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green" title="E88">A65</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7_1_4_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_7_1_4_1_1_3_1"/>
( <font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">q3</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 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E26:35_7_1_4_1_1"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_1_3_1_1"><b>case </b></a><a NAME="E1:35_7_1_4_1_1_3_1_1"/>
<font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E89">A66</font></i>: <font color="Maroon" title="c26">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<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_5.html#E25:35_7_1_4_1_1"><i><font color="Green" title="E87">A64</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>
;<br/><a NAME="E3:35_7_1_4_1_1_3_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_7_1_4_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E54:35"><i><font color="Green" title="E64">A41</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_7_1_4_1_1_3_1_1"><i><font color="Green" title="E89">A66</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_1_3_1_2"><b>case </b></a><a NAME="E1:35_7_1_4_1_1_3_1_2"/>
<font color="Maroon" title="c26">q3</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
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E89">A67</font></i>: <font color="Maroon" title="c26">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><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_5.html#E25:35_7_1_4_1_1"><i><font color="Green" title="E87">A64</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>
;<br/><a NAME="E3:35_7_1_4_1_1_3_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_7_1_4_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E54:35"><i><font color="Green" title="E64">A41</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_7_1_4_1_1_3_1_2"><i><font color="Green" title="E89">A67</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E28:35_7_1_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_2"><b>case </b></a><a NAME="E1:35_7_1_4_1_2"/><i><font color="Green" title="E79">A68</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c22">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c18">t</font>,<font color="Maroon" title="c21">r</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_7_1"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E13:35_7_1"><i><font color="Green" title="E77">A54</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">B0</font> = <font color="Maroon" title="c22">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c24">g</font> = <font color="Maroon" title="c16">g0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c23">B0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c23">B0</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:35_7_1_4_1_2"/>
<font color="Maroon" title="c16">g0</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E53:35"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jgraph_5.html#E56:35"><i><font color="Green" title="E65">A42</font></i></a>, <a class="ref" href="jgraph_5.html#T10" target="_self" title="JGRAPH_5:th.10">Th10</a></i>;<br/><a NAME="E6:35_7_1_4_1_2"/><b>then </b><i><font color="Green" title="E80">A69</font></i>: 
<font color="Maroon" title="c24">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E7:35_7_1_4_1_2"/><i><font color="Green" title="E81">A70</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c18">t</font>,<font color="Maroon" title="c21">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_2"><i><font color="Green" title="E79">A68</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/><a NAME="E8:35_7_1_4_1_2"/>
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_2"><i><font color="Green" title="E79">A68</font></i></a></i>;<br/><a NAME="E9:35_7_1_4_1_2"/><b>then </b><i><font color="Green" title="E82">A71</font></i>: 
<font color="Maroon" title="c21">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:35_7_1_4_1_2"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_2"><i><font color="Green" title="E79">A68</font></i></a></i>;<br/><a NAME="E11:35_7_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:35_7_1_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <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="c19">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">r</font> &amp; <font color="Maroon" title="c17">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_7_1"><i><font color="Green" title="E72">A49</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_7_1"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_7_1_4_1_2"><i><font color="Green" title="E82">A71</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:35_7_1_4_1_2"/><i><font color="Green" title="E83">A72</font></i>: 
( <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">r1</font> &amp; <font color="Maroon" title="c25">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">r</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_7_1_4_1_2"><i><font color="Green" title="E79">A68</font></i></a>, <a class="txt" href="jgraph_5.html#E6:35_7_1_4_1_2"><i><font color="Green" title="E80">A69</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_7_1_4_1_2"><i><font color="Green" title="E81">A70</font></i></a>, <a class="ref" href="topreal5.html#T14" title="TOPREAL5:th.14">TOPREAL5:14</a></i>;<br/><a NAME="E15:35_7_1_4_1_2"/>
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c18">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a></i>;<br/><a NAME="E16:35_7_1_4_1_2"/><b>then </b><i><font color="Green" title="E84">A73</font></i>: 
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:35_7_1_4_1_2"/>
<font color="Maroon" title="c25">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:35_7_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r2</font> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <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_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a></i>;<br/><a NAME="E19:35_7_1_4_1_2"/><b>then </b><i><font color="Green" title="E85">A74</font></i>: 
<font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:35_7_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:35_7_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E58:35"><i><font color="Green" title="E67">A44</font></i></a></i>;<br/><a NAME="E22:35_7_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E33:35"><i><font color="Green" title="E54">A31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">q3</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="E24:35_7_1_4_1_2"/><i><font color="Green" title="E86">A75</font></i>: 
( <font color="Maroon" title="c26">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c26">q3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/><i><font color="Green" title="E87">A76</font></i>: <font color="Maroon" title="c26">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<font color="Maroon" title="c15">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_2"><i><font color="Green" title="E86">A75</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_2"><i><font color="Green" title="E85">A74</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35_7_1_4_1_2"><i><font color="Green" title="E84">A73</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green" title="E88">A77</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_2"><i><font color="Green" title="E86">A75</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_7_1_4_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_7_1_4_1_2_3_1"/>
( <font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c26">q3</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 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E26:35_7_1_4_1_2"><i><font color="Green" title="E88">A77</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_2_3_1_1"><b>case </b></a><a NAME="E1:35_7_1_4_1_2_3_1_1"/>
<font color="Maroon" title="c26">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E89">A78</font></i>: <font color="Maroon" title="c26">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<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_5.html#E25:35_7_1_4_1_2"><i><font color="Green" title="E87">A76</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>
;<br/><a NAME="E3:35_7_1_4_1_2_3_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_7_1_4_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E54:35"><i><font color="Green" title="E64">A41</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_7_1"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_2"><i><font color="Green" title="E85">A74</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_2"><i><font color="Green" title="E86">A75</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_7_1_4_1_2_3_1_1"><i><font color="Green" title="E89">A78</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_2_3_1_2"><b>case </b></a><a NAME="E1:35_7_1_4_1_2_3_1_2"/>
<font color="Maroon" title="c26">q3</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
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E89">A79</font></i>: <font color="Maroon" title="c26">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><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_5.html#E25:35_7_1_4_1_2"><i><font color="Green" title="E87">A76</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>
;<br/><a NAME="E3:35_7_1_4_1_2_3_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_7_1"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_7_1_4_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E41:35"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jgraph_5.html#E54:35"><i><font color="Green" title="E64">A41</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1_4_1_2"><i><font color="Green" title="E83">A72</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_7_1_4_1_2"><i><font color="Green" title="E85">A74</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_7_1_4_1_2"><i><font color="Green" title="E86">A75</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_7_1_4_1_2_3_1_2"><i><font color="Green" title="E89">A79</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E28:35_7_1_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_7_1_4_1_3"><b>case </b></a><a NAME="E1:35_7_1_4_1_3"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_7_1_4_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_7"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_7_1"><i><font color="Green" title="E72">A49</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_7_1"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_7_1"><i><font color="Green" title="E78">A55</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:35_7_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_7"/>
for <font color="Olive" title="b1">q</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>  st <font color="Olive" title="b1">q</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="c1">P</font> holds <br/><font color="Olive" title="b1">q</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><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c17">R</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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>consider </b><font color="Maroon" title="c18">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">R</font></span>)</span><b> such that </b><br/><a NAME="E67:35"/><i><font color="Green" title="E71">A80</font></i>: 
( <font color="Maroon" title="c18">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:35"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E68:35"/><i><font color="Green" title="E72">A81</font></i>: 
<font color="Maroon" title="c18">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E69:35"/><i><font color="Green" title="E73">A82</font></i>: 
<font color="Maroon" title="c18">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E70:35"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> holds  <font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font>
 
;<br/>
<a NAME="E71:35"/><b>then </b><i><font color="Green" title="E74">A83</font></i>: 
<font color="Maroon" title="c15">h2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_5.html#T35" target="_self" title="JGRAPH_5:th.35">Th35</a></i>;<br/>
<a NAME="E72:35"/><i><font color="Green" title="E75">A84</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E73:35"/><i><font color="Green" title="E76">A85</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">R</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E74:35"/><b>then </b><i><font color="Green" title="E77">A86</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E75:35"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c15">h2</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="E76:35"/><b>then </b><i><font color="Green" title="E78">A87</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E72:35"><i><font color="Green" title="E75">A84</font></i></a>, <a class="txt" href="jgraph_5.html#E74:35"><i><font color="Green" title="E77">A86</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E77:35"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">g1</font> = <font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E76:35"><i><font color="Green" title="E78">A87</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E79:35"/><i><font color="Green" title="E79">A88</font></i>: 
(  ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) implies for <font color="Olive" title="b1">q</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>  st <font color="Olive" title="b1">q</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="c1">P</font> holds <br/><font color="Olive" title="b1">q</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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_8"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive" title="b2">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">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>, <font color="Maroon" title="c21">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:35_8"/><i><font color="Green" title="E80">A89</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">t</font> &amp; <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p</font> &amp; <font color="Maroon" title="c20">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_8_1"/>
 ex <font color="Olive" title="b1">q</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> st <br/>( <font color="Olive" title="b1">q</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="c1">P</font> &amp; <font color="Olive" title="b1">q</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/><b>then consider </b><font color="Maroon" title="c22">q</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="E3:35_8_1"/><i><font color="Green" title="E81">A90</font></i>: 
( <font color="Maroon" title="c22">q</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="c1">P</font> &amp; <font color="Maroon" title="c22">q</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/> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">R</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c17">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon" title="c23">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_8_1"/><i><font color="Green" title="E82">A91</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> &amp; <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8_1"><i><font color="Green" title="E81">A90</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:35_8_1"/><i><font color="Green" title="E83">A92</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:35_8_1"/><b>then </b>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <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_5.html#E6:35_8_1"><i><font color="Green" title="E82">A91</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:35_8_1"/><i><font color="Green" title="E84">A93</font></i>: 
( <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><i><font color="Green" title="E85">A94</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">r</font> = 
<font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">q</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_8_1"><i><font color="Green" title="E82">A91</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c22">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><a NAME="E12:35_8_1"/>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> <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_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a></i>;<br/><a NAME="E13:35_8_1"/><b>then </b><i><font color="Green" title="E86">A95</font></i>: 
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E87">A96</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t</font> = 
<font color="Maroon" title="c15">h2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">p</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c20">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_8_1_4_1"/>
( <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">t</font> or <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> or <font color="Maroon" title="c21">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_1"><b>case </b></a><a NAME="E1:35_8_1_4_1_1"/><i><font color="Green" title="E88">A97</font></i>: 
<font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">t</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c25">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c24">r</font>,<font color="Maroon" title="c21">t</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_8_1"><i><font color="Green" title="E82">A91</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E13:35_8_1"><i><font color="Green" title="E86">A95</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">B0</font> = <font color="Maroon" title="c25">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c27">g</font> = <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">B0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">B0</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:35_8_1_4_1_1"/>
<font color="Maroon" title="c19">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E68:35"><i><font color="Green" title="E72">A81</font></i></a>, <a class="txt" href="jgraph_5.html#E71:35"><i><font color="Green" title="E74">A83</font></i></a>, <a class="ref" href="jgraph_5.html#T10" target="_self" title="JGRAPH_5:th.10">Th10</a></i>;<br/><a NAME="E6:35_8_1_4_1_1"/><b>then </b><i><font color="Green" title="E89">A98</font></i>: 
<font color="Maroon" title="c27">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E7:35_8_1_4_1_1"/><i><font color="Green" title="E90">A99</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c24">r</font>,<font color="Maroon" title="c21">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_1"><i><font color="Green" title="E88">A97</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/><a NAME="E8:35_8_1_4_1_1"/>
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_1"><i><font color="Green" title="E88">A97</font></i></a></i>;<br/><a NAME="E9:35_8_1_4_1_1"/><b>then </b><i><font color="Green" title="E91">A100</font></i>: 
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:35_8_1_4_1_1"/>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_1"><i><font color="Green" title="E88">A97</font></i></a></i>;<br/><a NAME="E11:35_8_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:35_8_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c22">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <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="c22">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c20">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_8_1"><i><font color="Green" title="E81">A90</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_8_1"><i><font color="Green" title="E85">A94</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1"><i><font color="Green" title="E87">A96</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_8_1_4_1_1"><i><font color="Green" title="E91">A100</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:35_8_1_4_1_1"/><i><font color="Green" title="E92">A101</font></i>: 
( <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">r1</font> &amp; <font color="Maroon" title="c28">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c21">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_1"><i><font color="Green" title="E88">A97</font></i></a>, <a class="txt" href="jgraph_5.html#E6:35_8_1_4_1_1"><i><font color="Green" title="E89">A98</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_8_1_4_1_1"><i><font color="Green" title="E90">A99</font></i></a>, <a class="ref" href="topreal5.html#T14" title="TOPREAL5:th.14">TOPREAL5:14</a></i>;<br/><a NAME="E15:35_8_1_4_1_1"/>
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c24">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">t</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a></i>;<br/><a NAME="E16:35_8_1_4_1_1"/><b>then </b><i><font color="Green" title="E93">A102</font></i>: 
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:35_8_1_4_1_1"/>
<font color="Maroon" title="c28">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:35_8_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r2</font> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <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_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a></i>;<br/><a NAME="E19:35_8_1_4_1_1"/><b>then </b><i><font color="Green" title="E94">A103</font></i>: 
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:35_8_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:35_8_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E73:35"><i><font color="Green" title="E76">A85</font></i></a></i>;<br/><a NAME="E22:35_8_1_4_1_1"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E34:35"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c29">q3</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="E24:35_8_1_4_1_1"/><i><font color="Green" title="E95">A104</font></i>: 
( <font color="Maroon" title="c29">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c29">q3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/><i><font color="Green" title="E96">A105</font></i>: <font color="Maroon" title="c29">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<font color="Maroon" title="c15">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_1"><i><font color="Green" title="E95">A104</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_1"><i><font color="Green" title="E94">A103</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35_8_1_4_1_1"><i><font color="Green" title="E93">A102</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green" title="E97">A106</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_1"><i><font color="Green" title="E95">A104</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8_1_4_1_1_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_8_1_4_1_1_3_1"/>
( <font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c29">q3</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 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E26:35_8_1_4_1_1"><i><font color="Green" title="E97">A106</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_1_3_1_1"><b>case </b></a><a NAME="E1:35_8_1_4_1_1_3_1_1"/>
<font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E98">A107</font></i>: <font color="Maroon" title="c29">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<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_5.html#E25:35_8_1_4_1_1"><i><font color="Green" title="E96">A105</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>
;<br/><a NAME="E3:35_8_1_4_1_1_3_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_8_1_4_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E69:35"><i><font color="Green" title="E73">A82</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_1"><i><font color="Green" title="E94">A103</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_1"><i><font color="Green" title="E95">A104</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_8_1_4_1_1_3_1_1"><i><font color="Green" title="E98">A107</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_1_3_1_2"><b>case </b></a><a NAME="E1:35_8_1_4_1_1_3_1_2"/>
<font color="Maroon" title="c29">q3</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
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E98">A108</font></i>: <font color="Maroon" title="c29">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><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_5.html#E25:35_8_1_4_1_1"><i><font color="Green" title="E96">A105</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>
;<br/><a NAME="E3:35_8_1_4_1_1_3_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_8_1_4_1_1_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E69:35"><i><font color="Green" title="E73">A82</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_1"><i><font color="Green" title="E92">A101</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_1"><i><font color="Green" title="E94">A103</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_1"><i><font color="Green" title="E95">A104</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_8_1_4_1_1_3_1_2"><i><font color="Green" title="E98">A108</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E28:35_8_1_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_2"><b>case </b></a><a NAME="E1:35_8_1_4_1_2"/><i><font color="Green" title="E88">A109</font></i>: 
<font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c25">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c21">t</font>,<font color="Maroon" title="c24">r</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jgraph_5.html#E6:35_8_1"><i><font color="Green" title="E82">A91</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E13:35_8_1"><i><font color="Green" title="E86">A95</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">B0</font> = <font color="Maroon" title="c25">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c27">g</font> = <font color="Maroon" title="c19">g1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c26">B0</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c26">B0</font></span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></i>;<br/><a NAME="E5:35_8_1_4_1_2"/>
<font color="Maroon" title="c19">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E68:35"><i><font color="Green" title="E72">A81</font></i></a>, <a class="txt" href="jgraph_5.html#E71:35"><i><font color="Green" title="E74">A83</font></i></a>, <a class="ref" href="jgraph_5.html#T10" target="_self" title="JGRAPH_5:th.10">Th10</a></i>;<br/><a NAME="E6:35_8_1_4_1_2"/><b>then </b><i><font color="Green" title="E89">A110</font></i>: 
<font color="Maroon" title="c27">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T10" title="TOPMETR:th.10">TOPMETR:10</a></i>;<br/><a NAME="E7:35_8_1_4_1_2"/><i><font color="Green" title="E90">A111</font></i>: 
 <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c21">t</font>,<font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_2"><i><font color="Green" title="E88">A109</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#T30" title="TOPMETR:th.30">TOPMETR:30</a></i>;<br/><a NAME="E8:35_8_1_4_1_2"/>
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_2"><i><font color="Green" title="E88">A109</font></i></a></i>;<br/><a NAME="E9:35_8_1_4_1_2"/><b>then </b><i><font color="Green" title="E91">A112</font></i>: 
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:35_8_1_4_1_2"/>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_2"><i><font color="Green" title="E88">A109</font></i></a></i>;<br/><a NAME="E11:35_8_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:35_8_1_4_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c22">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <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="c22">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">r</font> &amp; <font color="Maroon" title="c20">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">t</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_8_1"><i><font color="Green" title="E81">A90</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_8_1"><i><font color="Green" title="E85">A94</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1"><i><font color="Green" title="E87">A96</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_8_1_4_1_2"><i><font color="Green" title="E91">A112</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">XREAL_1:134</a></i>;<br/><b>then consider </b><font color="Maroon" title="c28">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E14:35_8_1_4_1_2"/><i><font color="Green" title="E92">A113</font></i>: 
( <font color="Maroon" title="c27">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">r1</font> &amp; <font color="Maroon" title="c28">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">r</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_8_1_4_1_2"><i><font color="Green" title="E88">A109</font></i></a>, <a class="txt" href="jgraph_5.html#E6:35_8_1_4_1_2"><i><font color="Green" title="E89">A110</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35_8_1_4_1_2"><i><font color="Green" title="E90">A111</font></i></a>, <a class="ref" href="topreal5.html#T14" title="TOPREAL5:th.14">TOPREAL5:14</a></i>;<br/><a NAME="E15:35_8_1_4_1_2"/>
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r4</font> where <font color="Olive" title="b1">r4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c21">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r4</font> &amp; <font color="Olive" title="b1">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a></i>;<br/><a NAME="E16:35_8_1_4_1_2"/><b>then </b><i><font color="Green" title="E93">A114</font></i>: 
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E17:35_8_1_4_1_2"/>
<font color="Maroon" title="c28">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:35_8_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r2</font> where <font color="Olive" title="b1">r2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r2</font> &amp; <font color="Olive" title="b1">r2</font> <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_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a></i>;<br/><a NAME="E19:35_8_1_4_1_2"/><b>then </b><i><font color="Green" title="E94">A115</font></i>: 
<font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:35_8_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:35_8_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">R</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E73:35"><i><font color="Green" title="E76">A85</font></i></a></i>;<br/><a NAME="E22:35_8_1_4_1_2"/><b>then </b>
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E34:35"><i><font color="Green" title="E55">A32</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c29">q3</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="E24:35_8_1_4_1_2"/><i><font color="Green" title="E95">A116</font></i>: 
( <font color="Maroon" title="c29">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c29">q3</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a></i>;<br/><i><font color="Green" title="E96">A117</font></i>: <font color="Maroon" title="c29">q3</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<font color="Maroon" title="c15">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_2"><i><font color="Green" title="E95">A116</font></i></a>, <a class="ref" href="pscomp_1.html#D29" title="PSCOMP_1:def.29">PSCOMP_1:def 29</a></i>
<br/>.= 
<font color="Maroon" title="c19">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">r1</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_2"><i><font color="Green" title="E94">A115</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35_8_1_4_1_2"><i><font color="Green" title="E93">A114</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>then </b><i><font color="Green" title="E97">A118</font></i>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_2"><i><font color="Green" title="E95">A116</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_8_1_4_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_8_1_4_1_2_3_1"/>
( <font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c29">q3</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 )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E26:35_8_1_4_1_2"><i><font color="Green" title="E97">A118</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_2_3_1_1"><b>case </b></a><a NAME="E1:35_8_1_4_1_2_3_1_1"/>
<font color="Maroon" title="c29">q3</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E98">A119</font></i>: <font color="Maroon" title="c29">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<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_5.html#E25:35_8_1_4_1_2"><i><font color="Green" title="E96">A117</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31" title="PSCOMP_1:func.31">E-max</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>
;<br/><a NAME="E3:35_8_1_4_1_2_3_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_8_1_4_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E69:35"><i><font color="Green" title="E73">A82</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_2"><i><font color="Green" title="E94">A115</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_2"><i><font color="Green" title="E95">A116</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_8_1_4_1_2_3_1_1"><i><font color="Green" title="E98">A119</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_2_3_1_2"><b>case </b></a><a NAME="E1:35_8_1_4_1_2_3_1_2"/>
<font color="Maroon" title="c29">q3</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
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E98">A120</font></i>: <font color="Maroon" title="c29">q3</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><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_5.html#E25:35_8_1_4_1_2"><i><font color="Green" title="E96">A117</font></i></a>, <a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>
;<br/><a NAME="E3:35_8_1_4_1_2_3_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35_8_1"><i><font color="Green" title="E83">A92</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><b>hence </b><a NAME="E4:35_8_1_4_1_2_3_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E69:35"><i><font color="Green" title="E73">A82</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_8_1"><i><font color="Green" title="E84">A93</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1_4_1_2"><i><font color="Green" title="E92">A113</font></i></a>, <a class="txt" href="jgraph_5.html#E19:35_8_1_4_1_2"><i><font color="Green" title="E94">A115</font></i></a>, <a class="txt" href="jgraph_5.html#E24:35_8_1_4_1_2"><i><font color="Green" title="E95">A116</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_8_1_4_1_2_3_1_2"><i><font color="Green" title="E98">A120</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E28:35_8_1_4_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_8_1_4_1_3"><b>case </b></a><a NAME="E1:35_8_1_4_1_3"/>
<font color="Maroon" title="c21">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_8_1_4_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_8"><i><font color="Green" title="E80">A89</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_8_1"><i><font color="Green" title="E81">A90</font></i></a>, <a class="txt" href="jgraph_5.html#E11:35_8_1"><i><font color="Green" title="E85">A94</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_8_1"><i><font color="Green" title="E87">A96</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E16:35_8_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_8"/>
for <font color="Olive" title="b1">q</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>  st <font color="Olive" title="b1">q</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="c1">P</font> holds <br/><font color="Olive" title="b1">q</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><b>end;</b></div>
<a NAME="E80:35"/><i><font color="Green" title="E80">A121</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x2</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="E1:35_9"/><i><font color="Green" title="E81">A122</font></i>: 
<font color="Maroon" title="c20">x2</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="c1">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c21">q3</font> = <font color="Maroon" title="c20">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:35_9"/>
<font color="Maroon" title="c21">q3</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> 
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E8:35"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="jgraph_5.html#E23:35"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jgraph_5.html#E31:35"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jgraph_5.html#E44:35"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="jgraph_5.html#E48:35"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jgraph_5.html#E49:35"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jgraph_5.html#E51:35"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E64:35"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_9"><i><font color="Green" title="E81">A122</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<b>hence </b><a NAME="E4:35_9"/>
<font color="Maroon" title="c20">x2</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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_5.html#E33:35"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jgraph_5.html#E1:35_9"><i><font color="Green" title="E81">A122</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E81:35"/>
<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="c1">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>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">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="E1:35_10"/>
<font color="Maroon" title="c20">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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">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="E3:35_10"/><i><font color="Green" title="E81">A123</font></i>: 
( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c21">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>  )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_10_1_1"/>
( <font color="Maroon" title="c21">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c21">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_10"><i><font color="Green" title="E81">A123</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_10_1_1_1"><b>case </b></a><a NAME="E1:35_10_1_1_1"/><i><font color="Green" title="E82">A124</font></i>: 
<font color="Maroon" title="c21">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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c22">p8</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="E3:35_10_1_1_1"/><i><font color="Green" title="E83">A125</font></i>: 
( <font color="Maroon" title="c22">p8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c22">p8</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_10"><i><font color="Green" title="E81">A123</font></i></a></i>;<br/><a NAME="E4:35_10_1_1_1"/><i><font color="Green" title="E84">A126</font></i>: 
<font color="Maroon" title="c21">p</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"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>;<br/>1 = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_10_1_1_1"><i><font color="Green" title="E83">A125</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></i>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_10_1_1_1"><i><font color="Green" title="E82">A124</font></i></a>, <a class="ref" href="complex1.html#T158" title="COMPLEX1:th.158">COMPLEX1:158</a></i>
;<br/><a NAME="E6:35_10_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="complex1.html#T161" title="COMPLEX1:th.161">COMPLEX1:161</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>;<br/><a NAME="E7:35_10_1_1_1"/><b>then </b>
( <font color="Maroon" title="c21">p</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">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> or <font color="Maroon" title="c21">p</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_5.html#E1:35_10_1_1_1"><i><font color="Green" title="E82">A124</font></i></a>, <a class="txt" href="jgraph_5.html#E4:35_10_1_1_1"><i><font color="Green" title="E84">A126</font></i></a>, <a class="ref" href="square_1.html#T110" title="SQUARE_1:th.110">SQUARE_1:110</a></i>;<br/><b>hence </b><a NAME="E8:35_10_1_1_1"/>
<font color="Maroon" title="c20">x</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="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E36:35"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="jgraph_5.html#E38:35"><i><font color="Green" title="E57">A34</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_10"><i><font color="Green" title="E81">A123</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_10_1_1_2"><b>case </b></a><a NAME="E1:35_10_1_1_2"/><i><font color="Green" title="E82">A127</font></i>: 
<font color="Maroon" title="c21">p</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>  <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="35_10_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_10_1_1_2_1"/>
not <font color="Maroon" title="c20">x</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="c1">P</font>
 ;<br/><a NAME="E2:35_10_1_1_2_1"/><b>then </b><i><font color="Green" title="E83">A128</font></i>: 
<font color="Maroon" title="c20">x</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="c1">P</font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E3:35_10"><i><font color="Green" title="E81">A123</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c17">R</font></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c17">R</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><b>then consider </b><font color="Maroon" title="c22">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_10_1_1_2_1"/><i><font color="Green" title="E84">A129</font></i>: 
( <font color="Maroon" title="c22">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x2</font> )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:35_10"><i><font color="Green" title="E81">A123</font></i></a>, <a class="txt" href="jgraph_5.html#E2:35_10_1_1_2_1"><i><font color="Green" title="E83">A128</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:35_10_1_1_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:35_10_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c22">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <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_5.html#E5:35_10_1_1_2_1"><i><font color="Green" title="E84">A129</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">t2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:35_10_1_1_2_1"/><i><font color="Green" title="E85">A130</font></i>: 
( <font color="Maroon" title="c22">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">t2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">t2</font> &amp; <font color="Maroon" title="c23">t2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div><i><font color="Green" title="E86">A131</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_10_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_10_1_1_2_1_2"/>
<font color="Maroon" title="c23">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:35_10_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c21">p</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">1,<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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E5:35_10_1_1_2_1"><i><font color="Green" title="E84">A129</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_10_1_1_2_1"><i><font color="Green" title="E85">A130</font></i></a>, <a class="ref" href="jgraph_5.html#T33" target="_self" title="JGRAPH_5:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:35_10_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_10_1_1_2"><i><font color="Green" title="E82">A127</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="35_10_1_1_2_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_10_1_1_2_1_3"/>
<font color="Maroon" title="c23">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:35_10_1_1_2_1_3"/><b>then </b>
<font color="Maroon" title="c21">p</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_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E67:35"><i><font color="Green" title="E71">A80</font></i></a>, <a class="txt" href="jgraph_5.html#E5:35_10_1_1_2_1"><i><font color="Green" title="E84">A129</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_10_1_1_2_1"><i><font color="Green" title="E85">A130</font></i></a>, <a class="ref" href="jgraph_5.html#T32" target="_self" title="JGRAPH_5:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E3:35_10_1_1_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_10_1_1_2"><i><font color="Green" title="E82">A127</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="E12:35_10_1_1_2_1"/><b>then </b><i><font color="Green" title="E87">A132</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">t2</font> &amp; <font color="Maroon" title="c23">t2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c21">p</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:35_10_1_1_2"><i><font color="Green" title="E82">A127</font></i></a>, <a class="txt" href="jgraph_5.html#E5:35_10_1_1_2_1"><i><font color="Green" title="E84">A129</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35_10_1_1_2_1"><i><font color="Green" title="E85">A130</font></i></a>, <a class="txt" href="jgraph_5.html#E10:35_10_1_1_2_1"><i><font color="Green" title="E86">A131</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E13:35_10_1_1_2_1"/><i><font color="Green" title="E88">A133</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="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/><a NAME="E14:35_10_1_1_2_1"/><i><font color="Green" title="E89">A134</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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/><b>then </b><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_5.html#E13:35_10_1_1_2_1"><i><font color="Green" title="E88">A133</font></i></a>, <a class="ref" href="jgraph_3.html#T10" title="JGRAPH_3:th.10">JGRAPH_3:10</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="square_1.html#T83" title="SQUARE_1:th.83">SQUARE_1:83</a></i>
;<br/><a NAME="E16:35_10_1_1_2_1"/><b>then </b><i><font color="Green" title="E90">A135</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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> <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</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> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_10_1_1_2_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_10_1_1_2_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_10_1_1_2_1_5_1"/>
( <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 </span><b>by </b><i><a class="txt" href="jgraph_5.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E16:35_10_1_1_2_1"><i><font color="Green" title="E90">A135</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_10_1_1_2_1_5_1_1"><b>case </b></a><a NAME="E1:35_10_1_1_2_1_5_1_1"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_10_1_1_2_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E7:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="jgraph_5.html#E8:35"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="jgraph_5.html#E9:35"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="jgraph_5.html#E23:35"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jgraph_5.html#E31:35"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jgraph_5.html#E44:35"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="jgraph_5.html#E48:35"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jgraph_5.html#E49:35"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jgraph_5.html#E51:35"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jgraph_5.html#E64:35"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_10_1_1_2_1"><i><font color="Green" title="E89">A134</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="35_10_1_1_2_1_5_1_2"><b>case </b></a><a NAME="E1:35_10_1_1_2_1_5_1_2"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_10_1_1_2_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E79:35"><i><font color="Green" title="E79">A88</font></i></a>, <a class="txt" href="jgraph_5.html#E12:35_10_1_1_2_1"><i><font color="Green" title="E87">A132</font></i></a>, <a class="txt" href="jgraph_5.html#E14:35_10_1_1_2_1"><i><font color="Green" title="E89">A134</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:35_10_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:35_10_1_1_2"/>
<font color="Maroon" title="c20">x</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="c1">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_10"/>
<font color="Maroon" title="c20">x</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="c1">P</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E82:35"/>
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">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="c1">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_5.html#E80:35"><i><font color="Green" title="E80">A121</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
