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

<span class="kw">let </span><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span><br/>  for <font color="Olive" title="b2">C</font> being    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b3">G</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Olive" title="b2">C</font>  st <font color="Olive" title="b1">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Olive" title="b1">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font> holds <br/><font color="Olive" title="b3">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">C</font> being    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c3">F</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon" title="c3">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c3">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font> holds <br/><font color="Olive" title="b2">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">C</font> be    <a href="rcomp_3.html#M1" title="RCOMP_3:mode.1">IntervalCover</a> of <font color="Maroon" title="c3">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c4">C</font>  st <font color="Maroon" title="c3">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c3">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font> holds <br/><font color="Olive" title="b1">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <a href="rcomp_3.html#M2" title="RCOMP_3:mode.2">IntervalCoverPts</a> of <font color="Maroon" title="c4">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c3">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  &amp; <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font> implies <font color="Maroon" title="c5">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:86"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
( <font color="Maroon" title="c3">F</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span> &amp; <font color="Maroon" title="c3">F</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="rcomp_3.html#V1" title="RCOMP_3:attr.1">connected</a>  )
 <span class="kw">and </span><br/><a NAME="E2:86"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c5">G</font> or  not <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c5">G</font> or <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> or  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c7">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c6">m</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:86"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c7">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c6">m</font>)</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font> implies <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E4:86"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:86_1"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:86_1"/><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">k</font>]
 <span class="kw">and </span><br/><a NAME="E3:86_1"/><span class="lab"><font color="Green" title="E69">A7</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E4:86_1"/><span class="lab"><font color="Green" title="E70">A8</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E5:86_1"/><span class="lab"><font color="Green" title="E71">A9</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<a NAME="E6:86_1"/><span class="lab"><font color="Green" title="E72">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E7:86_1"/><span class="lab"><font color="Green" title="E73">A11</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86_1"><span class="lab"><font color="Green" title="E71">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:86_1"/>
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E9:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A12</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:86_1"/><span class="lab"><font color="Green" title="E75">A13</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:86_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:86_1_1"/>
( ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> ) or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86_1"><span class="lab"><font color="Green" title="E76">A14</font></span></a>, <a class="txt" href="#E7:86_1"><span class="lab"><font color="Green" title="E73">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:86_1_1_1"/><span class="lab"><font color="Green" title="E77">A15</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">and </span><br/><a NAME="E2:86_1_1_1"/><span class="lab"><font color="Green" title="E78">A16</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E3:86_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:86_1_1_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/><a NAME="E4:86_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A17</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></span>;<br/><a NAME="E5:86_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="txt" href="#E2:86_1_1_1"><span class="lab"><font color="Green" title="E78">A16</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:86_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:86_1_1_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="rcomp_3.html#T64" target="_self" title="RCOMP_3:th.64">Th78</a></span>;<br/><a NAME="E7:86_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1_1_1"><span class="lab"><font color="Green" title="E79">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:86_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a>, <a class="txt" href="#E4:86_1"><span class="lab"><font color="Green" title="E70">A8</font></span></a>, <a class="txt" href="#E10:86_1"><span class="lab"><font color="Green" title="E75">A13</font></span></a>, <a class="txt" href="#E9:86_1"><span class="lab"><font color="Green" title="E74">A12</font></span></a>, <a class="txt" href="#E1:86_1_1_1"><span class="lab"><font color="Green" title="E77">A15</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_2"/><span class="lab"><font color="Green" title="E77">A18</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:86_1_1_2"/><span class="lab"><font color="Green" title="E78">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#T51" target="_self" title="RCOMP_3:th.51">Th65</a></span>;<br/><a NAME="E3:86_1_1_2"/><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E1:86_1_1_2"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:86_1_1_2_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_2"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A21</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:86_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:86_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A22</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#T63" target="_self" title="RCOMP_3:th.63">Th77</a></span>;<br/><a NAME="E4:86_1_1_2_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:86_1_1_2_1_1"><span class="lab"><font color="Green" title="E80">A21</font></span></a>, <a class="ref" href="rcomp_3.html#T64" target="_self" title="RCOMP_3:th.64">Th78</a></span>;<br/><a NAME="E5:86_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1_1_2_1_1"><span class="lab"><font color="Green" title="E81">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:86_1_1_2_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E1:86_1_1_2"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E3:86_1_1_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_2_1_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:86_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#T61" target="_self" title="RCOMP_3:th.61">Th75</a></span>;<br/><a NAME="E3:86_1_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> &amp; <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E4:86_1_1_2_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="txt" href="#E1:86_1_1_2"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E3:86_1_1_2"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_3"/><span class="lab"><font color="Green" title="E77">A23</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:86_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A24</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:86_1_1_3_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font> or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_3_1_1"/><span class="lab"><font color="Green" title="E79">A25</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">z</font> = <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/><a NAME="E2:86_1_1_3_1_1"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:86_1"><span class="lab"><font color="Green" title="E72">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E3:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A26</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E4:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86_1"><span class="lab"><font color="Green" title="E69">A7</font></span></a>, <a class="txt" href="#E1:86_1_1_3"><span class="lab"><font color="Green" title="E77">A23</font></span></a></span>;<br/><a NAME="E5:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A28</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:86_1_1_3_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E79">A25</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A29</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E8:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A30</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E9:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A31</font></span>: 
 not <font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E5:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E82">A28</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/><a NAME="E10:86_1_1_3_1_1"/><span class="lab"><font color="Green" title="E86">A32</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="rcomp_3.html#D2" target="_self" title="RCOMP_3:def.2">Def2</a></span>;<br/><a NAME="E11:86_1_1_3_1_1"/><span class="lab"><font color="Green" title="E87">A33</font></span>: 
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E84">A30</font></span></a>, <a class="txt" href="#E5:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E82">A28</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E12:86_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E13:86_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E86">A32</font></span></a></span>;<br/><a NAME="E14:86_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E87">A33</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E15:86_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> (<font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font>)</span>)</span>
 ;<br/><a NAME="E16:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A34</font></span>: 
<font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>;<br/><a NAME="E17:86_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> is  <a href="xxreal_2.html#V4" title="XXREAL_2:attr.4">bounded_above</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#T43" title="XXREAL_2:th.43">XXREAL_2:43</a></span>;<br/><a NAME="E18:86_1_1_3_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E88">A34</font></span></a>, <a class="txt" href="#E9:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E85">A31</font></span></a>, <a class="ref" href="rcomp_3.html#T2" target="_self" title="RCOMP_3:th.2">Th2</a></span>;<br/><a NAME="E19:86_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A35</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/><a NAME="E20:86_1_1_3_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E3:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E80">A26</font></span></a>, <a class="txt" href="#E7:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E83">A29</font></span></a>, <a class="txt" href="#E4:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E81">A27</font></span></a>, <a class="ref" href="rcomp_3.html#T62" target="_self" title="RCOMP_3:th.62">Th76</a></span>;<br/><span class="kw">hence </span><a NAME="E21:86_1_1_3_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1_3"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="txt" href="#E19:86_1_1_3_1_1"><span class="lab"><font color="Green" title="E89">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:86_1_1_3_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:86_1_1_3_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:86"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E2:86_1_1_3"><span class="lab"><font color="Green" title="E78">A24</font></span></a>, <a class="ref" href="rcomp_3.html#D3" target="_self" title="RCOMP_3:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:86"/><span class="lab"><font color="Green" title="E67">A36</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E6:86"/>
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E5:86"><span class="lab"><font color="Green" title="E67">A36</font></span></a>, <a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E66">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:86"/>
 not <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c7">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="seq_1.html#K1" title="SEQ_1:func.1">K623</a>(<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c6">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:86"><span class="lab"><font color="Green" title="E65">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
