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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:20"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:20"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:20"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> is <a href="uniform1.html#V1" target="_self">uniformly_continuous</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:20"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:20"/><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 <b>and </b><br/><a NAME="E10:20"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<a NAME="E11:20"/><i><font color="Green">E62</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>
<a NAME="E12:20"/><i><font color="Green">E80</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E13:20"/><b>then </b><i><font color="Green">E82</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>
<a NAME="E14:20"/><i><font color="Green">E83</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E15:20"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">u2</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">u2</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/>

<a NAME="E2:20_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:20"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E17:20"/><b>then </b><i><font color="Green">E87</font></i>: 
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T201">REAL_2:201</a></i>;<br/>
<a NAME="E18:20"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<a NAME="E19:20"/>
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T80">INT_1:80</a></i>;<br/>
<a NAME="E21:20"/><i><font color="Green">E90</font></i>: 
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E22:20"/><i><font color="Green">E92</font></i>: 
2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E23:20"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="uniform1.html#E2:20"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E24:20"/><b>then </b><i><font color="Green">E94</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E25:20"/><b>then </b><i><font color="Green">E95</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E26:20"/><i><font color="Green">E97</font></i>: 
4 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="uniform1.html#T2" target="_self">Th2</a>, <a class="txt" href="uniform1.html#E4:20"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E27:20"/>
<span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="uniform1.html#E2:20"><i><font color="Green">E36</font></i></a>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E28:20"/><b>then </b>
1 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:20"/><b>then </b><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E30:20"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E31:20"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E33:20"/><i><font color="Green">E101</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="ref" href="uniform1.html#D1" target="_self">Def1</a>, );<br/></i>
<i><font color="Green">E102</font></i>: <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1 = 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
1

;<br/>
<a NAME="E35:20"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E36:20"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E37:20"/>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="int_1.html#D5">INT_1:def 5</a></i>;<br/>
<a NAME="E38:20"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E39:20"/><b>then </b>
<span class="p1"><a href="int_1.html#K2">[/</a><span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
;<br/>
<a NAME="E40:20"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E41:20"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>;<br/>
<a NAME="E42:20"/>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E43:20"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E44:20"/><b>then </b><i><font color="Green">E109</font></i>: 
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3"><a href="int_1.html#K2">[/</a><span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K2">\]</a></span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E45:20"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font>, <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( <font color="Olive">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r1</font> <a href="real_1.html#K5">-</a> <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:20_3"/><i><font color="Green">E113</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E2:20_3"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:20_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:20_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:20_3"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:20_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:20_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:20_3"/>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:20_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E10:20_3"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E6:20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uniform1.html#E7:20"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>

<b>thus </b><a NAME="E11:20_3"/>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="uniform1.html#E6:20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uniform1.html#E7:20"><i><font color="Green">E58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E46:20"/><i><font color="Green">E122</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:20_4"/><i><font color="Green">E124</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:20_4"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:20_4"/>
<font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:20_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:20_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i>, , <a class="ref" href="uniform1.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:20_4"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>
<b>hence </b><a NAME="E7:20_4"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 <b>by </b><i>, , <a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E47:20"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Olive">r1</font> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:20_5"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:20_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:20_5"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_5"/>
<font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E48:20"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_6"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:20_6"/><i><font color="Green">E129</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E6:20_6"/><i><font color="Green">E130</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> &amp; <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> )
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E131</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E8:20_6"/><b>then </b><i><font color="Green">E132</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> &amp; <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_6_2_1"/>
( <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">nx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_2_1_1"><b>case </b></a><a NAME="E1:20_6_2_1_1"/>
<font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:20_6_2_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:20_6_2_1_1"/><b>then </b>
<font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="uniform1.html#E9:20"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:20_6_2_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:20_6_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, <a class="txt" href="uniform1.html#E6:20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><b>hence </b><a NAME="E6:20_6_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_6_2_1_2"><b>case </b></a><a NAME="E1:20_6_2_1_2"/>
<font color="Maroon">nx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:20_6_2_1_2"/><b>then </b>
<font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="uniform1.html#E6:20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uniform1.html#E7:20"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:20_6_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:20_6_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:20_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<i><font color="Green">E135</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E51:20"/><i><font color="Green">E164</font></i>: 
4 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E52:20"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="uniform1.html#T3" target="_self">Th3</a>, , </i>;<br/>
<a NAME="E53:20"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<a NAME="E54:20"/><i><font color="Green">E167</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_8"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:20_8"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E6:20_8"/><i><font color="Green">E170</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> &amp; <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:20_8"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ry</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E9:20_8"/><i><font color="Green">E173</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ry</font>
 
<b>by </b><i>, <a class="txt" href="uniform1.html#E11:20"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E10:20_8"/><i><font color="Green">E174</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T125">REAL_2:125</a></i>;<br/>
<a NAME="E11:20_8"/>
<font color="Maroon">nx</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="uniform1.html#E11:20"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:20_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">nx</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>
<a NAME="E13:20_8"/><b>then </b>
1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E14:20_8"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">rs</font> where <font color="Olive">rs</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">rs</font> &amp; <font color="Olive">rs</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a>, , <a class="ref" href="uniform1.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E15:20_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E55:20"/><i><font color="Green">E176</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E56:20"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E57:20"/><b>then </b><i><font color="Green">E177</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E58:20"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_9"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:20_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:20"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E60:20"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E10:20"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E61:20"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="uniform1.html#E9:20"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E62:20"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E180</font></i>: the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
<b>by </b><i><a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>
;<br/>
<a NAME="E64:20"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> holds <br/><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_11"/><i><font color="Green">E182</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 ;<br/>

<a NAME="E2:20_11"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:20_11"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:20_11"/><i><font color="Green">E185</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_11_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_11_1_1_1"><b>case </b></a><a NAME="E1:20_11_1_1_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:20_11_1_1_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:20_11_1_1_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:20_11_1_1_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:20_11_1_1_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="uniform1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:20_11_1_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="uniform1.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><b>hence </b><a NAME="E7:20_11_1_1_1"/>
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_11_1_1_2"><b>case </b></a><a NAME="E1:20_11_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:20_11_1_1_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><i><font color="Green">E192</font></i>: <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="uniform1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
;<br/><b>hence </b><a NAME="E5:20_11_1_1_2"/>
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_11"/>
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E65:20"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">h1</font> is <a href="uniform1.html#V2" target="_self">decreasing</a>
 
<b>by </b><i/>;<br/>
<a NAME="E66:20"/><i><font color="Green">E194</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_12"/><i><font color="Green">E195</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 ;<br/>

<a NAME="E2:20_12"/><b>then </b><i><font color="Green">E196</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:20_12"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:20_12"/><i><font color="Green">E198</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_12_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_12_1_1_1"><b>case </b></a><a NAME="E1:20_12_1_1_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:20_12_1_1_1"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E3:20_12_1_1_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:20_12_1_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:20_12_1_1_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:20_12_1_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><b>hence </b><a NAME="E7:20_12_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_12_1_1_2"><b>case </b></a><a NAME="E1:20_12_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:20_12_1_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="uniform1.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><i><font color="Green">E205</font></i>: <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
0
<b>by </b><i>, </i>
;<br/><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="uniform1.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>
;<br/><b>hence </b><a NAME="E5:20_12_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:20_12"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E67:20"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">Q</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:20_13"/><i><font color="Green">E206</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> )
 ;<br/>

<a NAME="E2:20_13"/><b>then </b>
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_13"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>;<br/>
<a NAME="E5:20_13"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="uniform1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E6:20_13"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:20_13"/><i><font color="Green">E210</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_13_1"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_13_1"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x3</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x3</font> = <font color="Maroon">x3</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i>, , <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>consider </b><font color="Maroon">y3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:20_13_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y3</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y3</font> = <font color="Maroon">y3</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i>, , <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E8:20_13_1"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y3</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r3</font> = <font color="Maroon">x3</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">y3</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E11:20_13_1"/><i><font color="Green">E217</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s3</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:20_13_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:20_13_1"/><b>then </b><i><font color="Green">E220</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s3</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:20_13_1"/><i><font color="Green">E221</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<a NAME="E15:20_13_1"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E16:20_13_1"/><b>then </b><i><font color="Green">E222</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s1</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:20_13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x3</font>
 
<b>by </b><i>, , <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E18:20_13_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:20_13"/><b>then </b>
<font color="Maroon">W</font> is <a href="tbsp_1.html#V6">bounded</a>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#D9">TBSP_1:def 9</a></i>;<br/>
<a NAME="E9:20_13"/><b>then </b>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , <a class="ref" href="tbsp_1.html#D10">TBSP_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E10:20_13"/>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E68:20"/>
 ex <font color="Olive">h</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">h</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">h</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">h</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; 5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">h</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Olive">h</font> is <a href="uniform1.html#V2" target="_self">decreasing</a> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">h</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">Q</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E5:20"><i><font color="Green">E55</font></i></a>, , <a class="txt" href="uniform1.html#E6:20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="uniform1.html#E7:20"><i><font color="Green">E58</font></i></a>, <a class="txt" href="uniform1.html#E11:20"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="uniform1.html#T5" target="_self">Th5</a>, </i>;<br/>


</div>
