<?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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c4">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="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="ref" href="topreal4.html#T3" title="TOPREAL4:th.3">TOPREAL4:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:31"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="topreal1.html#NR2" title="TOPREAL1:NR.2">is_S-Seq</a>  &amp; <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="topreal4.html#D1" title="TOPREAL4:def.1">TOPREAL4:def 1</a></i>;<br/>
<a NAME="E5:31"/>
( <font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> &amp; <font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> &amp; <font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/>
<a NAME="E6:31"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E31">A5</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:31"/><i><font color="Green" title="E32">A6</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c7">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:31_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 ;<br/>

<b>hence </b><a NAME="E2:31_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:31"/><i><font color="Green" title="E34">A8</font></i>: 
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/>
<a NAME="E13:31"/><i><font color="Green" title="E35">A9</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:31"/><b>then </b><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:31"/><i><font color="Green" title="E37">A11</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:31"/><i><font color="Green" title="E38">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> $1</span>)</span> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>;<br/>
<a NAME="E17:31"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E15:31"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:31"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E19:31"/><b>then </b><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1]
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="jct_misc.html#T7" title="JCT_MISC:th.7">JCT_MISC:7</a></i>;<br/>
<a NAME="E20:31"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<a NAME="E21:31"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="jordan20.html#E20:31"><i><font color="Green" title="E40">A14</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c9">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E23:31"/><i><font color="Green" title="E41">A15</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 ;<br/>
<a NAME="E24:31"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/>
<a NAME="E25:31"/><b>then </b><i><font color="Green" title="E42">A16</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:31"/><i><font color="Green" title="E43">A17</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E27:31"/>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:31"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/>
<a NAME="E28:31"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E26:31"><i><font color="Green" title="E43">A17</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E29:31"/><b>then </b>
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E30:31"/><b>then </b><i><font color="Green" title="E44">A18</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E31:31"/><b>then </b><i><font color="Green" title="E45">A19</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E32:31"/><b>then </b><i><font color="Green" title="E46">A20</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E25:31"><i><font color="Green" title="E42">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:31"/><b>then </b><i><font color="Green" title="E47">A21</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">pk</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> 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="E35:31"/><i><font color="Green" title="E48">A22</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E30:31"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E36:31"/><i><font color="Green" title="E49">A23</font></i>: 
<font color="Maroon" title="c10">pk</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">kn</font> = <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1"/>
( <font color="Maroon" title="c10">pk</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c10">pk</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E36:31"><i><font color="Green" title="E49">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>case </b></a><a NAME="E1:31_2_1_1"/><i><font color="Green" title="E50">A24</font></i>: 
<font color="Maroon" title="c10">pk</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_1"/><i><font color="Green" title="E51">A25</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c12">p44</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>;<br/><i><font color="Green" title="E52">A26</font></i>: <font color="Maroon" title="c10">pk</font> = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_1"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="binarith.html#T58" title="BINARITH:th.58">BINARITH:58</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E3:31_2_1_1_1_1_1"/><i><font color="Green" title="E53">A27</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:31_2_1_1_1_1_1"/><i><font color="Green" title="E54">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E6:31_2_1_1_1_1_1"/><i><font color="Green" title="E55">A29</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E7:31_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E8:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E56">A30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_1"><i><font color="Green" title="E55">A29</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E9:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E57">A31</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_1"><i><font color="Green" title="E54">A28</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1_1_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E11:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E58">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A33</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:31_2_1_1_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_1"><i><font color="Green" title="E54">A28</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_1"><i><font color="Green" title="E56">A30</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E60">A34</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31_2_1_1_1_1_1"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c14">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E16:31_2_1_1_1_1_1"/>
<font color="Maroon" title="c14">ia</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E16:31"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E61">A35</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:31_2_1_1_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_1"><i><font color="Green" title="E57">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E19:31_2_1_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A36</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E11:31_2_1_1_1_1_1"><i><font color="Green" title="E58">A32</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E17:31_2_1_1_1_1_1"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,1
<b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_1"><i><font color="Green" title="E57">A31</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_1"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_1_1_1_1"><i><font color="Green" title="E62">A36</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E22:31_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_1"><i><font color="Green" title="E57">A31</font></i></a></i>;<br/><a NAME="E23:31_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E24:31_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E25:31_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E63">A37</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="sprect_4.html#T4" title="SPRECT_4:th.4">SPRECT_4:4</a></i>;<br/><a NAME="E26:31_2_1_1_1_1_1"/>
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_1_1_1_1_5"/><i><font color="Green" title="E64">A38</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_1_1_1_1_5"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:31_2_1_1_1_1_1_5"/><b>then </b><i><font color="Green" title="E65">A39</font></i>: 
<font color="Maroon" title="c15">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_1_5"><i><font color="Green" title="E64">A38</font></i></a></i>;<br/>
<a NAME="E4:31_2_1_1_1_1_1_5"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:31_2_1_1_1_1_1_5"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_1"><i><font color="Green" title="E53">A27</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:31_2_1_1_1_1_1_5"/><b>then </b><i><font color="Green" title="E66">A40</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E10:31"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1_1_5_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_1_5_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_1_5_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_1_5_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_1_1_1_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_1_5"><i><font color="Green" title="E66">A40</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_1_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_1_5"><i><font color="Green" title="E65">A39</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_1_5_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_1_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_1"><i><font color="Green" title="E52">A26</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="E8:31_2_1_1_1_1_1_5"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E27:31_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_1_1_1_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="jordan20.html#D1" target="_self" title="JORDAN20:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A42</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c12">p44</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>;<br/><a NAME="E4:31_2_1_1_1_1_2"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E5:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E54">A44</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E55">A45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_1_1_1_2_1"/><i><font color="Green" title="E56">A46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:31_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">p9</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">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span> holds <br/><font color="Olive" title="b1">p9</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_1"><i><font color="Green" title="E56">A46</font></i></a>, <a class="ref" href="binarith.html#T58" title="BINARITH:th.58">BINARITH:58</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_2"><i><font color="Green" title="E51">A41</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_2_1_1_1_1_2"/><i><font color="Green" title="E56">A47</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:31_2_1_1_1_1_2"/><i><font color="Green" title="E57">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:31_2_1_1_1_1_2"/><i><font color="Green" title="E58">A49</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E11:31_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E12:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E10:31_2_1_1_1_1_2"><i><font color="Green" title="E58">A49</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E60">A51</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:31_2_1_1_1_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E15:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E61">A52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E62">A53</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:31_2_1_1_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2"><i><font color="Green" title="E57">A48</font></i></a>, <a class="txt" href="jordan20.html#E12:31_2_1_1_1_1_2"><i><font color="Green" title="E59">A50</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A54</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E16:31_2_1_1_1_1_2"><i><font color="Green" title="E62">A53</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c14">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E20:31_2_1_1_1_1_2"/>
<font color="Maroon" title="c14">ia</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E16:31"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E21:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E64">A55</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:31_2_1_1_1_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31_2_1_1_1_1_2"><i><font color="Green" title="E60">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E23:31_2_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E65">A56</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2"><i><font color="Green" title="E61">A52</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:31_2_1_1_1_1_2"><i><font color="Green" title="E64">A55</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,1
<b>by </b><i><a class="txt" href="jordan20.html#E13:31_2_1_1_1_1_2"><i><font color="Green" title="E60">A51</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_1_1_1_2"><i><font color="Green" title="E63">A54</font></i></a>, <a class="txt" href="jordan20.html#E24:31_2_1_1_1_1_2"><i><font color="Green" title="E65">A56</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E26:31_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31_2_1_1_1_1_2"><i><font color="Green" title="E60">A51</font></i></a></i>;<br/><a NAME="E27:31_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E28:31_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E29:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A57</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="sprect_4.html#T4" title="SPRECT_4:th.4">SPRECT_4:4</a></i>;<br/><a NAME="E30:31_2_1_1_1_1_2"/><i><font color="Green" title="E67">A58</font></i>: 
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_1_1_1_2_5"/><i><font color="Green" title="E68">A59</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_1_1_1_2_5"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:31_2_1_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E69">A60</font></i>: 
<font color="Maroon" title="c15">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_5"><i><font color="Green" title="E68">A59</font></i></a></i>;<br/>
<a NAME="E4:31_2_1_1_1_1_2_5"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:31_2_1_1_1_1_2_5"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E7:31_2_1_1_1_1_2"><i><font color="Green" title="E56">A47</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:31_2_1_1_1_1_2_5"/><b>then </b><i><font color="Green" title="E70">A61</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E10:31"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1_2_5_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_5_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_2_5_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_2_5_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E29:31_2_1_1_1_1_2"><i><font color="Green" title="E66">A57</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_5"><i><font color="Green" title="E70">A61</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_1_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2"><i><font color="Green" title="E55">A45</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_5"><i><font color="Green" title="E69">A60</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_5_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_2_5_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="jordan20.html#T1" target="_self" title="JORDAN20:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_1_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2"><i><font color="Green" title="E55">A45</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_5"><i><font color="Green" title="E69">A60</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:31_2_1_1_1_1_2_5"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c15">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,1,<font color="Maroon" title="c8">i</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E32:31_2_1_1_1_1_2"/><i><font color="Green" title="E68">A62</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E15:31"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E33:31_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E34:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E69">A63</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E35:31_2_1_1_1_1_2"/><i><font color="Green" title="E70">A64</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E36:31_2_1_1_1_1_2"/><i><font color="Green" title="E71">A65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_1_1_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="jordan20.html#E32:31_2_1_1_1_1_2"><i><font color="Green" title="E68">A62</font></i></a>, <a class="txt" href="jordan20.html#E34:31_2_1_1_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E37:31_2_1_1_1_1_2"/><i><font color="Green" title="E72">A66</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:31"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E38:31_2_1_1_1_1_2"/><i><font color="Green" title="E73">A67</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E39:31_2_1_1_1_1_2"/><i><font color="Green" title="E74">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_1_1_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E41:31_2_1_1_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E32:31_2_1_1_1_1_2"><i><font color="Green" title="E68">A62</font></i></a>, <a class="txt" href="jordan20.html#E34:31_2_1_1_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="txt" href="jordan20.html#E35:31_2_1_1_1_1_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E42:31_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E75">A69</font></i>: <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E38:31_2_1_1_1_1_2"><i><font color="Green" title="E73">A67</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E32:31"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E44:31_2_1_1_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E36:31_2_1_1_1_1_2"><i><font color="Green" title="E71">A65</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E45:31_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E32:31_2_1_1_1_1_2"><i><font color="Green" title="E68">A62</font></i></a>, <a class="txt" href="jordan20.html#E34:31_2_1_1_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="txt" href="jordan20.html#E37:31_2_1_1_1_1_2"><i><font color="Green" title="E72">A66</font></i></a>, <a class="txt" href="jordan20.html#E39:31_2_1_1_1_1_2"><i><font color="Green" title="E74">A68</font></i></a>, <a class="txt" href="jordan20.html#E43:31_2_1_1_1_1_2"><i><font color="Green" title="E75">A69</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E46:31_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E47:31_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,1,<font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E48:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E76">A70</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="sprect_3.html#T33" title="SPRECT_3:th.33">SPRECT_3:33</a></i>;<br/><a NAME="E49:31_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E77">A71</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E29:31_2_1_1_1_1_2"><i><font color="Green" title="E66">A57</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E50:31_2_1_1_1_1_2"/><i><font color="Green" title="E78">A72</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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="jordan20.html#E48:31_2_1_1_1_1_2"><i><font color="Green" title="E76">A70</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green" title="E79">A73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_1_1_1_2_7"/>
 ex <font color="Olive" title="b1">p51</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/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c17">p51</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="E3:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E80">A74</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E4:31_2_1_1_1_1_2_7"/>
<font color="Maroon" title="c17">p51</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="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">Y3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E81">A75</font></i>: 
( <font color="Maroon" title="c17">p51</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y3</font> &amp; <font color="Maroon" title="c18">Y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i5</font></span>)</span> where <font color="Olive" title="b1">i5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i5</font> &amp; <font color="Olive" title="b1">i5</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">kk</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E82">A76</font></i>: 
( <font color="Maroon" title="c18">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a></i>;<br/><a NAME="E9:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E83">A77</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1_1_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:31_2_1_1_1_1_2_7"/><b>then </b><i><font color="Green" title="E84">A78</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:31_2_1_1_1_1_2_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:31_2_1_1_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:31_2_1_1_1_1_2_7"/><b>then </b><i><font color="Green" title="E85">A79</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E86">A80</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:31_2_1_1_1_1_2_7"/><i><font color="Green" title="E87">A81</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="jordan5c.html#T26" title="JORDAN5C:th.26">JORDAN5C:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1_2_7_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> or ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7"><i><font color="Green" title="E86">A80</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self" title="JORDAN20:th.2">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E88">A82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c20">k2</font> = <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font>;<br/><a NAME="E2:31_2_1_1_1_1_2_7_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="jordan5c.html#T25" title="JORDAN5C:th.25">JORDAN5C:25</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E89">A83</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_1_1_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2_1_1_1_1_2_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_2_7_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2_7"><i><font color="Green" title="E83">A77</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_1_1_1_2_7_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2"><i><font color="Green" title="E55">A45</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A82</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E89">A83</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E5:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E90">A84</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E6:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a></i>;<br/><a NAME="E7:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E91">A85</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E8:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E92">A86</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A84</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E9:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E93">A87</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E91">A85</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1_1_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="binarith.html#T68" title="BINARITH:th.68">BINARITH:68</a></i>;<br/><a NAME="E11:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font>]
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A82</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A84</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A86</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E93">A87</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E12:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/><a NAME="E13:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E14:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E94">A88</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E25:31"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan20.html#E35:31"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A84</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A86</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E15:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E95">A89</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E32:31"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E96">A90</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_1_1_1_2_7"><i><font color="Green" title="E87">A81</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E17:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E97">A91</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A84</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A86</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E94">A88</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E18:31_2_1_1_1_1_2_7_1_1_1"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Olive" title="b1">p8</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E19:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E98">A92</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A84</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A86</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E94">A88</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E96">A90</font></i></a></i>;<br/><a NAME="E20:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E99">A93</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</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="E1:31_2_1_1_1_1_2_7_1_1_1_2"/><i><font color="Green" title="E100">A94</font></i>: 
<font color="Maroon" title="c21">z</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:31_2_1_1_1_1_2_7_1_1_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i7</font></span>)</span> where <font color="Olive" title="b1">i7</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i7</font> &amp; <font color="Olive" title="b1">i7</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7"><i><font color="Green" title="E86">A80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:31_2_1_1_1_1_2_7_1_1_1_2"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_1_2"><i><font color="Green" title="E100">A94</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E100">A95</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E22:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E101">A96</font></i>: 
<font color="Maroon" title="c6">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E23:31_2_1_1_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E24:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E102">A97</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_1_1_1_2_7"><i><font color="Green" title="E84">A78</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7"><i><font color="Green" title="E85">A79</font></i></a>, <a class="txt" href="jordan20.html#E22:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E101">A96</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E25:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E103">A98</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_1_1_1_2_7"><i><font color="Green" title="E84">A78</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E26:31_2_1_1_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7"><i><font color="Green" title="E85">A79</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E27:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E24:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E102">A97</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E103">A98</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E28:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E104">A99</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E17:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E97">A91</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E99">A93</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><a NAME="E29:31_2_1_1_1_1_2_7_1_1_1"/><i><font color="Green" title="E105">A100</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E100">A95</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><a NAME="E30:31_2_1_1_1_1_2_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A82</font></i></a>, <a class="txt" href="jordan20.html#E19:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E98">A92</font></i></a>, <a class="txt" href="jordan20.html#E28:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E104">A99</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self" title="JORDAN20:th.3">Th3</a></i>;<br/><a NAME="E31:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A82</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E32:31_2_1_1_1_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2_7"><i><font color="Green" title="E83">A77</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7"><i><font color="Green" title="E86">A80</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E95">A89</font></i></a>, <a class="txt" href="jordan20.html#E19:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E98">A92</font></i></a>, <a class="txt" href="jordan20.html#E28:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E104">A99</font></i></a>, <a class="txt" href="jordan20.html#E29:31_2_1_1_1_1_2_7_1_1_1"><i><font color="Green" title="E105">A100</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self" title="JORDAN20:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E33:31_2_1_1_1_1_2_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E88">A101</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E89">A102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c20">k2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E3:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E90">A103</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</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="E1:31_2_1_1_1_1_2_7_1_1_2_1"/><i><font color="Green" title="E91">A104</font></i>: 
<font color="Maroon" title="c21">z</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:31_2_1_1_1_1_2_7_1_1_2_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i7</font></span>)</span> where <font color="Olive" title="b1">i7</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i7</font> &amp; <font color="Olive" title="b1">i7</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7"><i><font color="Green" title="E86">A80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:31_2_1_1_1_1_2_7_1_1_2_1"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_2_1"><i><font color="Green" title="E91">A104</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E91">A105</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Olive" title="b1">p8</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E5:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E92">A106</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="jordan5c.html#T23" title="JORDAN5C:th.23">JORDAN5C:23</a></i>;<br/><a NAME="E6:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E93">A107</font></i>: 
<font color="Maroon" title="c6">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E7:31_2_1_1_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E94">A108</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_1_1_1_2_7"><i><font color="Green" title="E84">A78</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7"><i><font color="Green" title="E85">A79</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E93">A107</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E9:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E95">A109</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_1_1_1_2_7"><i><font color="Green" title="E84">A78</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:31_2_1_1_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7"><i><font color="Green" title="E85">A79</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E94">A108</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E95">A109</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E12:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E96">A110</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E90">A103</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E92">A106</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><a NAME="E13:31_2_1_1_1_1_2_7_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="jordan5c.html#T25" title="JORDAN5C:th.25">JORDAN5C:25</a></i>;<br/><a NAME="E14:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E97">A111</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green" title="E98">A112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_7_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_1_1_1_2_7_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2_1_1_1_1_2_7_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:31_2_1_1_1_1_2_7_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E4:31_2_1_1_1_1_2_7_1_1_2_2"/><b>then </b><i><font color="Green" title="E99">A113</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E91">A105</font></i></a>, <a class="txt" href="jordan20.html#E12:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E96">A110</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E97">A111</font></i></a></i>;<br/><a NAME="E5:31_2_1_1_1_1_2_7_1_1_2_2"/><i><font color="Green" title="E100">A114</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E6:31_2_1_1_1_1_2_7_1_1_2_2"/>
<font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E7:31_2_1_1_1_1_2_7_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E8:31_2_1_1_1_1_2_7_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2"><i><font color="Green" title="E55">A45</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_1_1_1_2_7"><i><font color="Green" title="E81">A75</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_1_1_1_2_7"><i><font color="Green" title="E82">A76</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_1_1_1_2_7"><i><font color="Green" title="E83">A77</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E89">A102</font></i></a>, <a class="txt" href="jordan20.html#E4:31_2_1_1_1_1_2_7_1_1_2_2"><i><font color="Green" title="E99">A113</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_1_1_1_2_7_1_1_2_2"><i><font color="Green" title="E100">A114</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self" title="JORDAN20:th.7">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E16:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E17:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E99">A115</font></i>: <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E15:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E98">A112</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/><a NAME="E19:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E100">A116</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E20:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E101">A117</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E18:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A115</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E21:31_2_1_1_1_1_2_7_1_1_2"/><i><font color="Green" title="E102">A118</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E19:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E100">A116</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:31_2_1_1_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="binarith.html#T68" title="BINARITH:th.68">BINARITH:68</a></i>;<br/><a NAME="E23:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1]
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E88">A101</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A115</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E101">A117</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E102">A118</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E24:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/><a NAME="E25:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E26:31_2_1_1_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E103">A119</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E25:31"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan20.html#E35:31"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A115</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E101">A117</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E102">A118</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E27:31_2_1_1_1_1_2_7_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_1_1_1_2_7"><i><font color="Green" title="E80">A74</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_1_1_1_2_7"><i><font color="Green" title="E87">A81</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E28:31_2_1_1_1_1_2_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E88">A101</font></i></a>, <a class="txt" href="jordan20.html#E26:31_2_1_1_1_1_2_7_1_1_2"><i><font color="Green" title="E103">A119</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:31_2_1_1_1_1_2_7"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E52:31_2_1_1_1_1_2"/>
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_1_1_1_2_8"/><i><font color="Green" title="E80">A120</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E81">A121</font></i>: 
<font color="Maroon" title="c17">p5</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="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_1_1_2_8_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E50:31_2_1_1_1_1_2"><i><font color="Green" title="E78">A72</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_1_1_1_2_8"><i><font color="Green" title="E81">A121</font></i></a>, <a class="ref" href="jordan20.html#T19" target="_self" title="JORDAN20:th.19">Th19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_8_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_8_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_1_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E51:31_2_1_1_1_1_2"><i><font color="Green" title="E79">A73</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_8"><i><font color="Green" title="E80">A120</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1_1_2_8_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_1_1_1_2_8_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_1_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:31_2_1_1_1_1_2"><i><font color="Green" title="E67">A58</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1_1_1_2_8"><i><font color="Green" title="E80">A120</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="E4:31_2_1_1_1_1_2_8"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E53:31_2_1_1_1_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="jordan20.html#E49:31_2_1_1_1_1_2"><i><font color="Green" title="E77">A71</font></i></a>, <a class="ref" href="jordan20.html#D1" target="_self" title="JORDAN20:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_2_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2"><b>case </b></a><a NAME="E1:31_2_1_2"/><i><font color="Green" title="E50">A122</font></i>: 
<font color="Maroon" title="c10">pk</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_2_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_1"/><i><font color="Green" title="E51">A123</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c12">p44</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>;<br/><i><font color="Green" title="E52">A124</font></i>: <font color="Maroon" title="c10">pk</font> = 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_1"><i><font color="Green" title="E51">A123</font></i></a>, <a class="ref" href="binarith.html#T58" title="BINARITH:th.58">BINARITH:58</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E3:31_2_1_2_1_1_1"/><i><font color="Green" title="E53">A125</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:31_2_1_2_1_1_1"/><i><font color="Green" title="E54">A126</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E6:31_2_1_2_1_1_1"/><i><font color="Green" title="E55">A127</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E7:31_2_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E8:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E56">A128</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_1"><i><font color="Green" title="E55">A127</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E9:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E57">A129</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_1"><i><font color="Green" title="E54">A126</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1_2_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E11:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E58">A130</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E59">A131</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:31_2_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_1"><i><font color="Green" title="E54">A126</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_1"><i><font color="Green" title="E56">A128</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:31_2_1_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E60">A132</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31_2_1_2_1_1_1"><i><font color="Green" title="E59">A131</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c14">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E17:31_2_1_2_1_1_1"/>
<font color="Maroon" title="c14">ia</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E16:31"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E61">A133</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:31_2_1_2_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_1"><i><font color="Green" title="E57">A129</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E20:31_2_1_2_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A134</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_1"><i><font color="Green" title="E58">A130</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E18:31_2_1_2_1_1_1"><i><font color="Green" title="E61">A133</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,1
<b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_1"><i><font color="Green" title="E57">A129</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_1"><i><font color="Green" title="E60">A132</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_2_1_1_1"><i><font color="Green" title="E62">A134</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E23:31_2_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_1"><i><font color="Green" title="E57">A129</font></i></a></i>;<br/><a NAME="E24:31_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E25:31_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E26:31_2_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E63">A135</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="sprect_4.html#T4" title="SPRECT_4:th.4">SPRECT_4:4</a></i>;<br/><a NAME="E27:31_2_1_2_1_1_1"/>
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_2_1_1_1_5"/><i><font color="Green" title="E64">A136</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_2_1_1_1_5"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:31_2_1_2_1_1_1_5"/><b>then </b><i><font color="Green" title="E65">A137</font></i>: 
<font color="Maroon" title="c15">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_1_5"><i><font color="Green" title="E64">A136</font></i></a></i>;<br/>
<a NAME="E4:31_2_1_2_1_1_1_5"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:31_2_1_2_1_1_1_5"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_1"><i><font color="Green" title="E53">A125</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:31_2_1_2_1_1_1_5"/><b>then </b><i><font color="Green" title="E66">A138</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E10:31"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_2_1_1_1_5_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_1_5_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_2_1_1_1_5_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_1_5_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E26:31_2_1_2_1_1_1"><i><font color="Green" title="E63">A135</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_1_5"><i><font color="Green" title="E66">A138</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_2_1_1_1_5_1_1_1"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_1"><i><font color="Green" title="E52">A124</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_1_5"><i><font color="Green" title="E65">A137</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_1_5_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_2_1_1_1_5_1_1_2"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_1"><i><font color="Green" title="E52">A124</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="E8:31_2_1_2_1_1_1_5"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E28:31_2_1_2_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_1"><i><font color="Green" title="E52">A124</font></i></a>, <a class="txt" href="jordan20.html#E26:31_2_1_2_1_1_1"><i><font color="Green" title="E63">A135</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self" title="JORDAN20:def.2">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E51">A139</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">A140</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>set </b><font color="Maroon" title="c12">p44</font> = <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>;<br/><a NAME="E4:31_2_1_2_1_1_2"/><i><font color="Green" title="E53">A141</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2"><i><font color="Green" title="E52">A140</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E5:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E54">A142</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><div><i><font color="Green" title="E55">A143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_2_1_1_2_1"/><i><font color="Green" title="E56">A144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><a NAME="E2:31_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_2_1"/><b>then </b>
for <font color="Olive" title="b1">p9</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">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 0</span>)</span> holds <br/><font color="Olive" title="b1">p9</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_1"><i><font color="Green" title="E56">A144</font></i></a>, <a class="ref" href="binarith.html#T58" title="BINARITH:th.58">BINARITH:58</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_2"><i><font color="Green" title="E51">A139</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E7:31_2_1_2_1_1_2"/><i><font color="Green" title="E56">A145</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:31_2_1_2_1_1_2"/><i><font color="Green" title="E57">A146</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:31_2_1_2_1_1_2"/><i><font color="Green" title="E58">A147</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E11:31_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E12:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E59">A148</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E10:31_2_1_2_1_1_2"><i><font color="Green" title="E58">A147</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E13:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A149</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2"><i><font color="Green" title="E57">A146</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:31_2_1_2_1_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E15:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E61">A150</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E62">A151</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:31_2_1_2_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2"><i><font color="Green" title="E57">A146</font></i></a>, <a class="txt" href="jordan20.html#E12:31_2_1_2_1_1_2"><i><font color="Green" title="E59">A148</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E18:31_2_1_2_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A152</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E16:31_2_1_2_1_1_2"><i><font color="Green" title="E62">A151</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="jordan20.html#E14:31"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c14">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E21:31_2_1_2_1_1_2"/>
<font color="Maroon" title="c14">ia</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E16:31"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E22:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E64">A153</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:31_2_1_2_1_1_2"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31_2_1_2_1_1_2"><i><font color="Green" title="E60">A149</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E24:31_2_1_2_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E65">A154</font></i>: <font color="Maroon" title="c13">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2"><i><font color="Green" title="E61">A150</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E22:31_2_1_2_1_1_2"><i><font color="Green" title="E64">A153</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/> <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,1
<b>by </b><i><a class="txt" href="jordan20.html#E13:31_2_1_2_1_1_2"><i><font color="Green" title="E60">A149</font></i></a>, <a class="txt" href="jordan20.html#E19:31_2_1_2_1_1_2"><i><font color="Green" title="E63">A152</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_2_1_1_2"><i><font color="Green" title="E65">A154</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><a NAME="E27:31_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31_2_1_2_1_1_2"><i><font color="Green" title="E60">A149</font></i></a></i>;<br/><a NAME="E28:31_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c13">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E29:31_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E30:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E66">A155</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="sprect_4.html#T4" title="SPRECT_4:th.4">SPRECT_4:4</a></i>;<br/><a NAME="E31:31_2_1_2_1_1_2"/><i><font color="Green" title="E67">A156</font></i>: 
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_2_1_1_2_5"/><i><font color="Green" title="E68">A157</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_2_1_1_2_5"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/>
<a NAME="E3:31_2_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E69">A158</font></i>: 
<font color="Maroon" title="c15">p5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_5"><i><font color="Green" title="E68">A157</font></i></a></i>;<br/>
<a NAME="E4:31_2_1_2_1_1_2_5"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<a NAME="E5:31_2_1_2_1_1_2_5"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E9:31"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E7:31_2_1_2_1_1_2"><i><font color="Green" title="E56">A145</font></i></a>, <a class="ref" href="topreal1.html#T12" title="TOPREAL1:th.12">TOPREAL1:12</a></i>;<br/>
<a NAME="E6:31_2_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E70">A159</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E10:31"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_2_1_1_2_5_1_1"/>
( <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_5_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_2_1_1_2_5_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_2_5_1_1_1"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E30:31_2_1_2_1_1_2"><i><font color="Green" title="E66">A155</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_5"><i><font color="Green" title="E70">A159</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_2_1_1_2_5_1_1_1"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2"><i><font color="Green" title="E55">A143</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_5"><i><font color="Green" title="E69">A158</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_5_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_2_1_1_2_5_1_1_2"/><b>then </b>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="jordan20.html#T1" target="_self" title="JORDAN20:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_2_1_1_2_5_1_1_2"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2"><i><font color="Green" title="E55">A143</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_5"><i><font color="Green" title="E69">A158</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:31_2_1_2_1_1_2_5"/>
<font color="Maroon" title="c15">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c15">g</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,1,<font color="Maroon" title="c8">i</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E33:31_2_1_2_1_1_2"/><i><font color="Green" title="E68">A160</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E15:31"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E34:31_2_1_2_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2"><i><font color="Green" title="E54">A142</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><a NAME="E35:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E69">A161</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E36:31_2_1_2_1_1_2"/><i><font color="Green" title="E70">A162</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2"><i><font color="Green" title="E52">A140</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a></i>;<br/><a NAME="E37:31_2_1_2_1_1_2"/><i><font color="Green" title="E71">A163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_2_1_1_2"><i><font color="Green" title="E53">A141</font></i></a>, <a class="txt" href="jordan20.html#E33:31_2_1_2_1_1_2"><i><font color="Green" title="E68">A160</font></i></a>, <a class="txt" href="jordan20.html#E35:31_2_1_2_1_1_2"><i><font color="Green" title="E69">A161</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E38:31_2_1_2_1_1_2"/><i><font color="Green" title="E72">A164</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:31"><i><font color="Green" title="E44">A18</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E39:31_2_1_2_1_1_2"/><i><font color="Green" title="E73">A165</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 <b>by </b><i><a class="txt" href="jordan20.html#E12:31"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2"><i><font color="Green" title="E52">A140</font></i></a>, <a class="ref" href="binarith.html#T69" title="BINARITH:th.69">BINARITH:69</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E40:31_2_1_2_1_1_2"/><i><font color="Green" title="E74">A166</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_2_1_1_2"><i><font color="Green" title="E53">A141</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">ia</font> = <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E42:31_2_1_2_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E33:31_2_1_2_1_1_2"><i><font color="Green" title="E68">A160</font></i></a>, <a class="txt" href="jordan20.html#E35:31_2_1_2_1_1_2"><i><font color="Green" title="E69">A161</font></i></a>, <a class="txt" href="jordan20.html#E36:31_2_1_2_1_1_2"><i><font color="Green" title="E70">A162</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E43:31_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E75">A167</font></i>: <font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E39:31_2_1_2_1_1_2"><i><font color="Green" title="E73">A165</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E32:31"><i><font color="Green" title="E46">A20</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E45:31_2_1_2_1_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31"><i><font color="Green" title="E45">A19</font></i></a>, <a class="txt" href="jordan20.html#E37:31_2_1_2_1_1_2"><i><font color="Green" title="E71">A163</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E46:31_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> &amp;  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E33:31_2_1_2_1_1_2"><i><font color="Green" title="E68">A160</font></i></a>, <a class="txt" href="jordan20.html#E35:31_2_1_2_1_1_2"><i><font color="Green" title="E69">A161</font></i></a>, <a class="txt" href="jordan20.html#E38:31_2_1_2_1_1_2"><i><font color="Green" title="E72">A164</font></i></a>, <a class="txt" href="jordan20.html#E40:31_2_1_2_1_1_2"><i><font color="Green" title="E74">A166</font></i></a>, <a class="txt" href="jordan20.html#E44:31_2_1_2_1_1_2"><i><font color="Green" title="E75">A167</font></i></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/><a NAME="E47:31_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c15">g</font>,<font color="Olive" title="b1">i2</font></span>)</span> where <font color="Olive" title="b1">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i2</font> &amp; <font color="Olive" title="b1">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E48:31_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">pk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">f</font>,1,<font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/><a NAME="E49:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E76">A168</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2"><i><font color="Green" title="E54">A142</font></i></a>, <a class="ref" href="sprect_3.html#T33" title="SPRECT_3:th.33">SPRECT_3:33</a></i>;<br/><a NAME="E50:31_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E77">A169</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E30:31_2_1_2_1_1_2"><i><font color="Green" title="E66">A155</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E51:31_2_1_2_1_1_2"/><i><font color="Green" title="E78">A170</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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="jordan20.html#E49:31_2_1_2_1_1_2"><i><font color="Green" title="E76">A168</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><div><i><font color="Green" title="E79">A171</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_2_1_1_2_7"/>
 ex <font color="Olive" title="b1">p51</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/>(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; not <font color="Olive" title="b1">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c17">p51</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="E3:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E80">A172</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><a NAME="E4:31_2_1_2_1_1_2_7"/>
<font color="Maroon" title="c17">p51</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="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">Y3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E81">A173</font></i>: 
( <font color="Maroon" title="c17">p51</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y3</font> &amp; <font color="Maroon" title="c18">Y3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i5</font></span>)</span> where <font color="Olive" title="b1">i5</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i5</font> &amp; <font color="Olive" title="b1">i5</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">kk</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E82">A174</font></i>: 
( <font color="Maroon" title="c18">Y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">kk</font> &amp; <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a></i>;<br/><a NAME="E9:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E83">A175</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:31_2_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E11:31_2_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E84">A176</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:31_2_1_2_1_1_2_7"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:31_2_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:31_2_1_2_1_1_2_7"/><b>then </b><i><font color="Green" title="E85">A177</font></i>: 
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E86">A178</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E16:31_2_1_2_1_1_2_7"/><i><font color="Green" title="E87">A179</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p51</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="jordan5c.html#T26" title="JORDAN5C:th.26">JORDAN5C:26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_2_1_1_2_7_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> or ( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2_7"><i><font color="Green" title="E86">A178</font></i></a>, <a class="ref" href="jordan20.html#T3" target="_self" title="JORDAN20:th.3">Th3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E88">A180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c20">k2</font> = <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font>;<br/><a NAME="E2:31_2_1_2_1_1_2_7_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="jordan5c.html#T25" title="JORDAN5C:th.25">JORDAN5C:25</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E89">A181</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_2_1_1_2_7_1_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2_1_2_1_1_2_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_2_7_1_1_1_1"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2"><i><font color="Green" title="E54">A142</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2_7"><i><font color="Green" title="E83">A175</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><b>hence </b><a NAME="E4:31_2_1_2_1_1_2_7_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2"><i><font color="Green" title="E55">A143</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A180</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E89">A181</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E90">A182</font></i>: <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> = 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c19">kk</font>

;<br/><a NAME="E6:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E91">A183</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E7:31_2_1_2_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="binarith.html#T68" title="BINARITH:th.68">BINARITH:68</a></i>;<br/><a NAME="E8:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font>]
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A180</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A182</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E91">A183</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E9:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/><a NAME="E10:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E11:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E92">A184</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E25:31"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan20.html#E35:31"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A182</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E91">A183</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E12:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E93">A185</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E32:31"><i><font color="Green" title="E46">A20</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E94">A186</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_2_1_1_2_7"><i><font color="Green" title="E87">A179</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E14:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E95">A187</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A182</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E91">A183</font></i></a>, <a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A184</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E15:31_2_1_2_1_1_2_7_1_1_1"/>
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Olive" title="b1">p8</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E16:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E96">A188</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E90">A182</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E91">A183</font></i></a>, <a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E92">A184</font></i></a>, <a class="txt" href="jordan20.html#E13:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E94">A186</font></i></a></i>;<br/><a NAME="E17:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E97">A189</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</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="E1:31_2_1_2_1_1_2_7_1_1_1_3"/><i><font color="Green" title="E98">A190</font></i>: 
<font color="Maroon" title="c21">z</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:31_2_1_2_1_1_2_7_1_1_1_3"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i7</font></span>)</span> where <font color="Olive" title="b1">i7</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i7</font> &amp; <font color="Olive" title="b1">i7</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2_7"><i><font color="Green" title="E86">A178</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:31_2_1_2_1_1_2_7_1_1_1_3"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_1_3"><i><font color="Green" title="E98">A190</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E98">A191</font></i>: 
<font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E19:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E99">A192</font></i>: 
<font color="Maroon" title="c6">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E20:31_2_1_2_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E21:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E100">A193</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7"><i><font color="Green" title="E84">A176</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7"><i><font color="Green" title="E85">A177</font></i></a>, <a class="txt" href="jordan20.html#E19:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E99">A192</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E22:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E101">A194</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7"><i><font color="Green" title="E84">A176</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E23:31_2_1_2_1_1_2_7_1_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7"><i><font color="Green" title="E85">A177</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E24:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E21:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E100">A193</font></i></a>, <a class="txt" href="jordan20.html#E22:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E101">A194</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E25:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E102">A195</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E95">A187</font></i></a>, <a class="txt" href="jordan20.html#E17:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E97">A189</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><a NAME="E26:31_2_1_2_1_1_2_7_1_1_1"/><i><font color="Green" title="E103">A196</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E98">A191</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><a NAME="E27:31_2_1_2_1_1_2_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A180</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E96">A188</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E102">A195</font></i></a>, <a class="ref" href="jordan20.html#T2" target="_self" title="JORDAN20:th.2">Th2</a></i>;<br/><a NAME="E28:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E88">A180</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E29:31_2_1_2_1_1_2_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">p51</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2_7"><i><font color="Green" title="E83">A175</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2_7"><i><font color="Green" title="E86">A178</font></i></a>, <a class="txt" href="jordan20.html#E12:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E93">A185</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E96">A188</font></i></a>, <a class="txt" href="jordan20.html#E25:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E102">A195</font></i></a>, <a class="txt" href="jordan20.html#E26:31_2_1_2_1_1_2_7_1_1_1"><i><font color="Green" title="E103">A196</font></i></a>, <a class="ref" href="jordan20.html#T7" target="_self" title="JORDAN20:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E30:31_2_1_2_1_1_2_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E88">A197</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font> )
 ;<br/><div class="add"><a NAME="E2:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E89">A198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>set </b><font color="Maroon" title="c20">k2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E3:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E90">A199</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">z</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="E1:31_2_1_2_1_1_2_7_1_1_2_1"/><i><font color="Green" title="E91">A200</font></i>: 
<font color="Maroon" title="c21">z</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:31_2_1_2_1_1_2_7_1_1_2_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Olive" title="b1">i7</font></span>)</span> where <font color="Olive" title="b1">i7</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i7</font> &amp; <font color="Olive" title="b1">i7</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2_7"><i><font color="Green" title="E86">A178</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:31_2_1_2_1_1_2_7_1_1_2_1"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:31"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_2_1"><i><font color="Green" title="E91">A200</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E91">A201</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p8</font> where <font color="Olive" title="b1">p8</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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Olive" title="b1">p8</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p8</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E5:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E92">A202</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="jordan5c.html#T23" title="JORDAN5C:th.23">JORDAN5C:23</a></i>;<br/><a NAME="E6:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E93">A203</font></i>: 
<font color="Maroon" title="c6">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="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E7:31_2_1_2_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c19">kk</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E94">A204</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7"><i><font color="Green" title="E84">A176</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7"><i><font color="Green" title="E85">A177</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E93">A203</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E9:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E95">A205</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E11:31_2_1_2_1_1_2_7"><i><font color="Green" title="E84">A176</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:31_2_1_2_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7"><i><font color="Green" title="E85">A177</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E94">A204</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E95">A205</font></i></a>, <a class="ref" href="topreal1.html#T15" title="TOPREAL1:th.15">TOPREAL1:15</a></i>;<br/><a NAME="E12:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E96">A206</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E90">A199</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E92">A202</font></i></a>, <a class="ref" href="jordan20.html#T28" target="_self" title="JORDAN20:th.28">Th28</a></i>;<br/><a NAME="E13:31_2_1_2_1_1_2_7_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c17">p51</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="jordan5c.html#T25" title="JORDAN5C:th.25">JORDAN5C:25</a></i>;<br/><a NAME="E14:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E97">A207</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><div><i><font color="Green" title="E98">A208</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_7_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_1_2_1_1_2_7_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><a NAME="E2:31_2_1_2_1_1_2_7_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:31_2_1_2_1_1_2_7_1_1_2_2"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2"><i><font color="Green" title="E54">A142</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E4:31_2_1_2_1_1_2_7_1_1_2_2"/><b>then </b><i><font color="Green" title="E99">A209</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</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">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E91">A201</font></i></a>, <a class="txt" href="jordan20.html#E12:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan20.html#E14:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E97">A207</font></i></a></i>;<br/><a NAME="E5:31_2_1_2_1_1_2_7_1_1_2_2"/><i><font color="Green" title="E100">A210</font></i>: 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>;<br/><a NAME="E6:31_2_1_2_1_1_2_7_1_1_2_2"/>
<font color="Maroon" title="c6">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="topreal1.html#D10" title="TOPREAL1:def.10">TOPREAL1:def 10</a></i>;<br/><a NAME="E7:31_2_1_2_1_1_2_7_1_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></i>;<br/><b>hence </b><a NAME="E8:31_2_1_2_1_1_2_7_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2"><i><font color="Green" title="E55">A143</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="txt" href="jordan20.html#E6:31_2_1_2_1_1_2_7"><i><font color="Green" title="E81">A173</font></i></a>, <a class="txt" href="jordan20.html#E8:31_2_1_2_1_1_2_7"><i><font color="Green" title="E82">A174</font></i></a>, <a class="txt" href="jordan20.html#E9:31_2_1_2_1_1_2_7"><i><font color="Green" title="E83">A175</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E89">A198</font></i></a>, <a class="txt" href="jordan20.html#E4:31_2_1_2_1_1_2_7_1_1_2_2"><i><font color="Green" title="E99">A209</font></i></a>, <a class="txt" href="jordan20.html#E5:31_2_1_2_1_1_2_7_1_1_2_2"><i><font color="Green" title="E100">A210</font></i></a>, <a class="ref" href="jordan20.html#T6" target="_self" title="JORDAN20:th.6">Th6</a></i>;<br/></div><b>end;</b></div><a NAME="E16:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E17:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><b>then </b><i><font color="Green" title="E99">A211</font></i>: <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E15:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E98">A208</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/><a NAME="E19:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E100">A212</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E20:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E101">A213</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E18:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A211</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E21:31_2_1_2_1_1_2_7_1_1_2"/><i><font color="Green" title="E102">A214</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E19:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E100">A212</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:31_2_1_2_1_1_2_7_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E13:31"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="binarith.html#T68" title="BINARITH:th.68">BINARITH:68</a></i>;<br/><a NAME="E23:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1]
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E88">A197</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A211</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E101">A213</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E102">A214</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E24:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E23:31"><i><font color="Green" title="E41">A15</font></i></a></i>;<br/><a NAME="E25:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c19">kk</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E26:31_2_1_2_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E103">A215</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E25:31"><i><font color="Green" title="E42">A16</font></i></a>, <a class="txt" href="jordan20.html#E35:31"><i><font color="Green" title="E48">A22</font></i></a>, <a class="txt" href="jordan20.html#E18:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E99">A211</font></i></a>, <a class="txt" href="jordan20.html#E20:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E101">A213</font></i></a>, <a class="txt" href="jordan20.html#E21:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E102">A214</font></i></a>, <a class="ref" href="jordan5c.html#T24" title="JORDAN5C:th.24">JORDAN5C:24</a></i>;<br/><a NAME="E27:31_2_1_2_1_1_2_7_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E3:31_2_1_2_1_1_2_7"><i><font color="Green" title="E80">A172</font></i></a>, <a class="txt" href="jordan20.html#E16:31_2_1_2_1_1_2_7"><i><font color="Green" title="E87">A179</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><b>hence </b><a NAME="E28:31_2_1_2_1_1_2_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:31"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="jordan20.html#E33:31"><i><font color="Green" title="E47">A21</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E88">A197</font></i></a>, <a class="txt" href="jordan20.html#E26:31_2_1_2_1_1_2_7_1_1_2"><i><font color="Green" title="E103">A215</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E18:31_2_1_2_1_1_2_7"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E53:31_2_1_2_1_1_2"/>
for <font color="Olive" title="b1">p5</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  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> holds <br/><font color="Olive" title="b1">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">p5</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_2_1_2_1_1_2_8"/><i><font color="Green" title="E80">A216</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">pk</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<a NAME="E2:31_2_1_2_1_1_2_8"/><b>then </b><i><font color="Green" title="E81">A217</font></i>: 
<font color="Maroon" title="c17">p5</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="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_2_1_1_2_8_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> or  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E51:31_2_1_2_1_1_2"><i><font color="Green" title="E78">A170</font></i></a>, <a class="txt" href="jordan20.html#E2:31_2_1_2_1_1_2_8"><i><font color="Green" title="E81">A217</font></i></a>, <a class="ref" href="jordan20.html#T19" target="_self" title="JORDAN20:th.19">Th19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_8_1_1_1"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_8_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_2_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E52:31_2_1_2_1_1_2"><i><font color="Green" title="E79">A171</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_8"><i><font color="Green" title="E80">A216</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_2_1_1_2_8_1_1_2"><b>case </b></a><a NAME="E1:31_2_1_2_1_1_2_8_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c17">p5</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:31_2_1_2_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E31:31_2_1_2_1_1_2"><i><font color="Green" title="E67">A156</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2_1_1_2_8"><i><font color="Green" title="E80">A216</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="E4:31_2_1_2_1_1_2_8"/>
<font color="Maroon" title="c17">p5</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E54:31_2_1_2_1_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:31"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:31"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="jordan20.html#E1:31_2_1_2"><i><font color="Green" title="E50">A122</font></i></a>, <a class="txt" href="jordan20.html#E50:31_2_1_2_1_1_2"><i><font color="Green" title="E77">A169</font></i></a>, <a class="ref" href="jordan20.html#D2" target="_self" title="JORDAN20:def.2">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:31_2_1_2"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E39:31"/>
( <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R1" title="JORDAN20:pred.1">is_Lin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> or <font color="Maroon" title="c4">p</font> <a href="jordan20.html#R2" title="JORDAN20:pred.2">is_Rin</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">e</font> )
 ;<br/>


</div>
